Time-dependent Gutzwiller theory of pairing fluctuations in the Hubbard model

Time-dependent Gutzwiller theory of pairing fluctuations in the Hubbard model

G. Seibold,1 _{F. Becca,}2_{and J. Lorenzana}3,4

1_{Institut für Physik, BTU Cottbus, P.O. Box 101344, 03013 Cottbus, Germany} 2_{CNR-INFM Democritos National Simulation Center, SISSA, I-34014 Trieste, Italy}

3_{Center for Statistical Mechanics and Complexity, INFM-CNR, Dipartimento di Fisica, Università di Roma “La Sapienza”,}

Piazzale Aldo Moro 2, 00185 Roma, Italy

4_{ISC-CNR, Via dei Taurini 19, 00185 Roma, Italy}

共Received 28 March 2008; published 23 July 2008兲

We present a method to compute pairing fluctuations on top of the Gutzwiller approximation共GA兲. Our investigations are based on a charge-rotational invariant GA energy functional which is expanded up to second order in the pair fluctuations. Equations of motion for the fluctuations lead to a renormalized ladder-type approximation. Both spectral functions and corrections to static quantities, such as the ground-state energy, are computed. The quality of the method is examined for the single-band Hubbard model, where we compare the dynamical pairing correlations for s- and d-wave symmetries with exact diagonalizations. We find a significant improvement with respect to analogous calculations done within the standard Hartree-Fock ladder approxima-tion. The technique has potential applications in the theory of Auger spectroscopy, superconductivity, and cold atom physics.

DOI:10.1103/PhysRevB.78.045114 PACS number共s兲: 71.10.Fd, 71.30.⫹h, 79.20.Fv I. INTRODUCTION

The present interest in the physics of strongly correlated fermion systems is accompanied by a “revival” of the Gutzwiller wave function 共GWF兲 共Ref. 1兲 in order to study

Hubbard-type models. In these Hamiltonians, doubly occu-pied sites contribute with an energy U to the total energy, though their weight is partially reduced by the Gutzwiller projector. Originally, the projector was applied to a Slater determinant describing a spatially uniform Fermi liquid. Along the years, states with long-range order have been con-sidered, including projected BCS wave functions,2–8_{used in} the context of high Tc.

Analytic evaluations of expectation values using the GWF are only possible in one9and infinite dimensions.10,11 In the latter limit, one recovers the so-called Gutzwiller approxima-tion共GA兲, first introduced by Gutzwiller himself in Ref.12. The GA, later on rederived as a saddle point within a slave-boson formulation of the Hubbard model,13_{yields an energy} functional for quasiparticles with renormalized hopping am-plitude. Therefore, it offers a simple and intuitive picture for the interplay between local correlation and electron kinetics. It is used in a variety of fields including the description of inhomogeneous states in cuprates,14 _{band-structure} calculations,15 _{and the theory of He}3_.16

In the past few years, two of us developed a scheme which allows the computation of Gaussian fluctuations on top of the GA.17–19 _{Within this so-called time-dependent} Gutzwiller approximation共TDGA兲, it is possible to evaluate dynamical correlation functions in the charge and spin chan-nel, which are in good agreement with exact diagonalization results over a wide parameter regime and constitute a signifi-cant improvement over the traditional Hartree-Fock plus random-phase approximation 共HF+RPA兲. The TDGA has been used in order to compute dynamical properties of inho-mogeneous states in cuprates. Results obtained in this way for the optical conductivity20_{and magnetic susceptibility}21,22

of stripe ordered phases have turned out to be in excellent agreement with experimental data.

In the present paper, we generalize the TDGA toward the inclusion of pairing fluctuations. The significance of such correlations is probably most prominent in the context of superconductivity, where the appearance of a singularity in the pair susceptibility signals the onset of Cooper-pair con-densation. Here, we aim to study the spectrum of pairing correlations in the normal state of the Hubbard model, an issue which has also been addressed, among others, within exact diagonalization23,24 and various Monte Carlo methods.25–31_{In the particle-hole channel, the coupling to an} electric or magnetic field yields the optical conductivity and magnetic susceptibility, respectively. In the particle-particle channel, direct measurements of pairing correlations are not so common since they do not couple to any classical field. However, principles for the measurement of the pair suscep-tibility in metals are discussed in Refs.32–35. The basic idea is to probe the fluctuating pair field of a metal in the normal state which is coupled to a superconductor via the tunnel current-voltage characteristics. In addition, the pairing corre-lation function for local pairs is the main ingredient in the theory of Auger spectroscopy.36–39 _{It also has relevance in} the field of ultracold atom physics.40

In materials which can be described within Hubbard-type models, a strong local repulsion induces so-called antibound states 共also known as two-particle resonances兲, above the two-particle continuum. The resonances appear as atomiclike features in the Auger spectrum. In Ref. 41we showed that, despite its computational simplicity, the TDGA yields an ex-cellent description of the two-particle response. In particular, it describes well the energy gap between band and antibound states and the relative spectral weight even far from the di-lute limit, in contrast to the “bare ladder approximation” 共BLA兲, which is restricted to the low-density regime.42,43 Thus, the TDGA allows one to extend ladder-type theories much beyond their supposed limit of validity. The result is an effective ladder theory where quasiparticles get heavier, as

usual due to correlations, and at the same time the effective interactions between quasiparticles become strongly renor-malized. These vertex and self-energy corrections are consis-tent with each other and do not suffer from the pitfalls fre-quently found in diagrammatic computations, where an improvement at the level of the self-energy alone leads to a degradation of the overall performance of the theory due to the lack of appropriate vertex corrections.38,39

In this paper, besides a thorough derivation of the “pairing TDGA,” we extend the approach to intersite correlations with extended s-wave and d-wave symmetries. We will show that the TDGA yields an effective interaction between qua-siparticles which is renormalized with respect to the bare U due to correlations but it does not include enough fluctuation effects to induce a superconducting instability. This is be-cause we start from a paramagnetic state treated in the GA, which does not have enough variational freedom to describe the scale J of magnetic fluctuations. In addition, within the RPA treatment for a paramagnet, the particle-particle and particle-hole channels are decoupled and do not influence each other. For spin-density wave ground states, the TDGA induces attractive interactions between nearest-neighbor pairs which are not present in the BLA approach based on HF. However, this is not enough to produce a superconduct-ing instability. Superconductivity due to an electronic mecha-nism requires the feedback of particle-hole fluctuations on the particle-particle channel and this goes beyond our ap-proach. On the other hand, at the level of spectroscopic quantities, which are our main concern here, this is a minor effect and our approach turns out to be in excellent agree-ment with exact diagonalization on a broad energy range.

Here and below we use the terms “ladder-type fluctuations,” “particle-particle RPA fluctuations,” and “pairing fluctuations” synonymously.

This paper is organized as follows: The formalism is pre-sented in detail in Sec. II, where as a first step we present the charge-rotational invariant GA functional from the Hubbard model in Sec. II A. Based on this functional, we show in Sec. II B how ladder-type fluctuations can be incorporated into the approach and how dynamical pair correlations can be computed. Results are presented in Sec. III, where we first exemplify the method by means of a two-site model. Then the dynamical pairing correlations for different symmetries are computed on small clusters and compared with exact diagonalization results and HF+ BLA computations. Finally, we conclude our investigations in Sec. IV.

II. FORMALISM A. Charge-rotational invariant GA The starting point is the one-band Hubbard model:

H =

兺

i,j,␴共tij −␮␦ij兲ci†␴cj_␴+ U

兺

i nˆi_↑nˆi_↓, 共1兲 where ci␴ 共ci␴

†_{兲 destroys 共creates兲 an electron with spin␴ at}

site i and nˆi␴= ci␴ †

ci␴. U is the on-site Hubbard repulsion, tij

denotes the hopping parameter between sites i and j, and␮is the chemical potential.

The Gutzwiller variational wave function can be written as

兩⌽G典 ⬅

兿

i

Pˆi兩␾典, 共2兲

where Pˆipartially projects out a doubly occupied state at site i from the uncorrelated wave function兩␾典. The latter one, in the traditional Gutzwiller approach,1_{is a Slater determinant} and the associated density matrix contains only the normal part:

␳ij␴␴⬘⬅ 具␾兩cj_␴⬘ †

ci␴兩␾典.

Here, we will consider a more general formulation in which 兩␾典 is a Hartree-Fock-Bogoliubov 共HFB兲 wave function and we define the anomalous part of the density matrix,

␬ij␴␴⬘⬅ 具␾兩cj␴⬘ci␴兩␾典.

In general ␳ and␬ can be considered as matrices in which the index i␴label columns and rows. It is also convenient to define a generalized density matrix,

R =

冉

␳ ␬

−␬ⴱ 1 −␳ⴱ

冊

. 共3兲

For a normal system, one can show that ␳ is the density matrix associated with a Slater determinant if and only if ␳2_{=␳. In the present case, the analogous condition for the}

Hartree-Fock-Bogoliubov wave function is given by the condition44–46

R2= R. 共4兲

In the general case, the normal part can describe charge-density wave and spin-charge-density wave broken symmetries. We will allow the ground state to have these broken symmetries, but we will assume it is normal so our saddle-point anoma-lous density matrix will vanish. We denote quantities at the saddle point as 0; thus␬0_{= 0. The anomalous part is}

impor-tant in order to compute fluctuations. Indeed, in the follow-ing, we consider an external time-dependent perturbation which induces pairing fluctuations and the system will be described by a time-dependent version of Eq.共2兲 with a 兩␾典

that has a BCS-type共i.e., Bogoliubov兲 form.

The charge-rotational invariant Gutzwiller functional for general charge and spin textures is derived in Appendix A. Essentially, one maps the superconducting problem into a problem without superconductivity by performing a rotation of a pseudospin vector in charge space.47At this point, one can derive the GA by using counting arguments6,16_{or infinite} dimension results10,11 _{or, as we do in Appendix A, by} ex-ploiting the well-known equivalence between the slave-boson method and the Gutzwiller approach.48–52 Alterna-tively, one can use a pure Gutzwiller formulation directly on the HFB wave function with an appropriate projector Pˆi.6,53

A judicious choice of Pˆi ensures that the local expectation

values are preserved by the projector: 具⌽G兩nˆi␴兩⌽G典

=具␾兩nˆi␴兩␾典 and 具⌽G兩ci↓ci↑兩⌽G典=具␾兩ci↓ci↑兩␾典. This provides a

The generalized energy functional for the Hubbard model reads F共␳,␬,D兲 =

兺

i,j tij具␾兩⌿i†Ai␶zAj⌿j兩␾典 −␮

兺

i ␳ii+ U

兺

i Di, 共5兲 which depends on the normal and anomalous parts of the density matrix and the variational double-occupancy param-eters Di. In Eq.共5兲,␳ii⬅兺␴␳ii␴␴and we have introduced the

Nambu notation

⌿i†=共ci†_↑,ci_↓兲, ⌿i=

冉

ci_↑ ci†_↓

冊

. 共6兲

It is also convenient to define the pseudospin vector Jជiwith

components Ji x =1 2⌿i †_␶ x⌿i= 1 2共ci↑ † ci↓ † + ci↓ci↑兲, 共7兲 Ji y =1 2⌿i †_␶ y⌿i= − i 2共ci↑ † ci↓ † − ci_↓ci_↑兲, 共8兲 Ji z =1 2⌿i †_␶ z⌿i= 1 2共ci↑ †_c i↑+ ci†_↓ci↓− 1兲, 共9兲

where ␶i denote the Pauli matrices. The components of Jជi

obey the standard commutation relations of a spin 1/2 alge-bra. We use boldface letters to indicate two-component vec-tors and 2⫻2 matrices in Nambu space, whereas we denote Cartesian vectors by using an arrow.

The raising and lowering operators are defined as Ji + = Ji x + iJi y = ci↑ † ci↓ † , 共10兲 Ji − = Ji x − iJi y = ci↓ci↑. 共11兲

Within these definitions the matrix Aiin Eq.共5兲 reads as

Ai=

冢

zi_↑+ zi_↓ 2 + zi_↑− zi_↓ 2 具Ji z_典 具Ji典 具Ji −_典 2具Ji典 关zi_↑− zi_↓兴 具Ji+典 2具Ji典 关zi↑− zi↓兴 zi↑+ zi↓ 2 − zi↑− zi↓ 2 具Ji z_典 具Ji典

冣

, 共12兲 and all expectation values 具...典 refer to the state 兩␾典. The Gutzwiller renormalization factors are given by

zi␴=

冑

Di−具Ji z_{典 − 具J} i典

冑

1/2 + 具Ji z_{典 − D} i+␴具Si z_{典 +}

冑

_{1/2 + 具J} i z_{典 − D} i−␴具Si z_典

冑

Di−具Ji z_{典 + 具J} i典

冑

1/4 − 共具Ji典 + 具Si z_典兲2 , 共13兲

with Ji⬅兩Jជi兩. Note that in the limit 具Ji⫾典=0, where the matrix Ai is diagonal, one recovers the standard Gutzwiller energy

functional as derived by Gebhard11 _{or Kotliar and} Ruckenstein:13 F共␳,D兲 =

兺

i,j tij␴zi␴zj␴␳␴␴ji −␮

兺

i ␳ii+ U

兺

i Di, 共14兲

with the hopping renormalization factors

zi␴=

冑

共␳ii␴␴− Di兲共1 −␳ii+ Di兲 +

冑

共␳ii␴¯␴¯− Di兲Di

冑

␳ii␴␴共1 −␳ii␴␴兲

. 共15兲 The minimization of the energy functional in Eq.共5兲 leads to

the stationary Gutzwiller wave function and to the associated stationary uncorrelated state兩␾₀典.

B. Calculation of pair fluctuations around general GA saddle points

The energy functional in Eq. 共5兲 is a convenient starting

point for the calculation of pair excitations on top of unre-stricted Gutzwiller wave functions. The main idea that was followed in Refs.17–19and41is to derive a time-dependent Gutzwiller approximation following the same approach used in nuclear physics to compute RPA fluctuations starting from

the Hartree-Fock-Bogoliubov mean field performed with an effective density-dependent potential.

In practice, one studies the response of the system to an external time-dependent perturbation, which in the present case induces small-amplitude oscillations in the particle-particle channel:

⌬共t兲 =

兺

ij

共fije−ı␻tci_↓cj_↑+ H.c.兲. 共16兲

The total energy function now acquires a new term, FT= F

+ F_⌬, with

F_⌬=具⌽G兩⌬兩⌽G典 =

兺

ij

f¯ije−ı␻t␬ji+ H.c., 共17兲

where f¯ij is the matrix element calculated in the GA. For

local matrix elements f¯ii= fii, whereas nonlocal matrix

ele-ments acquire renormalization factors, i.e., f¯ij= fijzizj. We

will come back to this issue below when we consider specific forms for ⌬. In this work, we will consider operators with antiparallel spins, as in Eq. 共16兲 so that only ␬ji↑↓ and its

conjugate appear in the equations. Therefore, for clarity, we can drop the spin indices in ␬.

In the following, we compute the time evolution of the density matrix by using equations of motion,45_{which depend}

only on variational energy functional 共5兲. In the limit of

small oscillations, the density-matrix elements will play the role of effective coordinates and energy functional共5兲 can be

expanded around the stationary solution up to second order in the density and double-occupancy deviations. We remind the reader that a subscript or superscript 0 indicates a quan-tity evaluated in the stationary state 兩␾0典 and that ␬0= 0.

Fluctuations are defined by the symbol ␦, e.g., ␦␳共t兲=␳共t兲 −␳0, except for␬for which the symbol␦is omitted, since its

mean-field value vanishes.

In the present case, particle-hole共ph兲 and particle-particle 共pp兲 sectors in the expansion are decoupled and one obtains F = F₀+ tr兵h0␦␳其 +␦Fph+␦Fpp, 共18兲 where we have introduced the Gutzwiller Hamiltonian:44–46

hij␴␴⬘关␳,D兴 = ⳵F ⳵␳ji␴⬘␴

␦␴,␴_⬘. 共19兲

This coincides with the Kotliar-Ruckenstein Hamiltonian matrix.13_{In particular, the diagonal elements共in the basis of} atomic orbitals兲 coincide with the Lagrange multipliers of the Kotliar-Ruckenstein method,⌺i_␴, after adding the

chemi-cal potential:

⌺i␴= ⳵F ⳵␳ii␴␴

+␮. 共20兲

In Ref.41we interpreted⌺i␴as a local GA self-energy.

Since we have included␮in Eq.共1兲, the eigenvalues␰of matrix共19兲 at the saddle point h0describe the single-particle excitations with respect to the chemical potential at the GA level. We denote the particle 共hole兲 energies above 共below兲 the Fermi energy by ␰_p␴ 共␰_h␴兲 with␰_p␴⬎0⬎␰_h␴.

The transformation from real space fermions ci␴ to GA

operators is written as ci␴=

兺

p ␾i␴共p兲ap␴+

兺

h ␾i␴共h兲ah␴. 共21兲

The amplitudes ␾i␴共p兲 and ␾i␴共h兲 correspond to the

eigen-functions of Eq. 共19兲 and the index p 共h兲 runs over empty

共occupied兲 states.

In the basis that diagonalizes the saddle-point Gutzwiller Hamiltonian h0_{, the second term in Eq.共18兲 can be written as}

tr兵h0_{␦␳其 =}

兺

p␴ ␰p␦␳pp

␴␴₊

兺

h␴ ␰h␦␳hh

␴␴_{. 共22兲}

This is apparently first order in the density matrix. However, because of the subsidiary condition in Eq. 共4兲, not all the

matrix elements of the generalized density matrix are inde-pendent. Using an analogous procedure as in Refs. 18 and

19, one can eliminate the diagonal matrix elements of the density matrix in term of ph matrix elements of␳and pp and hh matrix elements of ␬, which in the limit of small-amplitude fluctuations can be taken as independent variables:45 tr兵h0_{␦␳其 =}

兺

ph␴␴_⬘ 共␰p−␰h兲␦␳ph␴␴⬘␦␳hp␴⬘␴+ 1 2

兺

pp⬘␴␴⬘ 共␰p+␰p⬘兲 ⫻兩␬pp⬘兩2− 1 2

兺

hh⬘␴␴⬘ 共␰h+␰h⬘兲兩␬hh⬘兩2. 共23兲

The first term together with ␦Fphcontributes to the particle-hole fluctuations and decouples from the pairing fluctuations so it can be omitted in the following. This part of the RPA problem was already studied in detail in Refs. 17 and 18, where it was shown that the ␦D fluctuations can be elimi-nated by assuming that they adjust instantaneously to the evolution of the density matrix共antiadiabaticity condition兲.

The particle-particle fluctuations are described by the sec-ond and third terms of Eq. 共23兲 and by the interaction part

␦Fpp. Note that, since␰p⬎0⬎␰h, Eq.共23兲 is a positive

defi-nite quadratic form in the effective coordinates, which cor-rectly ensures the stability in the absence of interactions. This would not be the case if the chemical potential was omitted in the functional.

Finally, the particle-particle part of the expansion reads ␦Fpp₌

兺

ijkl

Vijkl␬ijⴱ␬kl, 共24兲

with Vijkl=共Vklij兲ⴱ. The matrix elements of the effective

in-teraction are given by

Viiii=

兺

j tij关共具cj↑ † ci↑典0+具ci↑ † cj↑典0兲Aj++ 0 A_i++

⬙

+共具cj↓ † ci↓典0 +具ci↓ † cj↓典0兲Aj−− 0 A_i−−

⬙

兴 + U 1 − ni , Viijj= − tijAi

⬘

A

⬘兵具c

j j↑ † ci↑典0+具cj↓ † ci↓典0其 for i ⫽ j, Vijjj=共Vjjij兲ⴱ= tijAi++0 A

⬘

j for i⫽ j, Vijii=共Viiij兲ⴱ= − tijA0j−−Ai

⬘

for i⫽ j,

Vijkl= 0 otherwise.

Here, Ai␶␶⬘共with␶,␶

⬘

=⫾兲 are the matrix elements of Eq.

共12兲 and we have defined the following abbreviations for the

derivatives: Ai

⬘

=

冏

⳵A_i+− ⳵具Ji −_典

冏

0 =

冏

⳵Ai−+ ⳵具Ji +_典

冏

0 , 共25兲 Ai

⬙

␶␶=

冏

⳵2_A i␶␶ ⳵具Ji+典⳵具Ji−典

冏

0 , 共26兲

where explicit expressions are given in Appendix B. It is interesting to observe that, in contrast to the charge excita-tions in the particle-hole channel, the evaluation of the pair excitations can be performed without any assumption on the time evolution of␦D. Only in the case of a superconducting

ground state, one would have a coupling between ph and pp fluctuations and, therefore, the necessity to invoke the antia-diabaticity condition in Refs. 17and18to eliminate the␦D deviations. Note also that, in contrast to HF theory 共where the expansion would be given by U兺i␦具Ji

+_典␦具J i

−_{典兲, Eq. 共} 24兲

contains correlations between distant pairs 共i.e., ␦具Ji+典␦具J−j典兲

and also processes where pairs are created and annihilated on neighboring sites共i.e.,␦具ci↑

† cj↓

†_典兲.

The remaining part of the formalism follows closely the particle-particle RPA as developed in nuclear physics,44,54 finite many-body systems,45 _{and the theory of} superconductivity.46,55 _{We also mention that more recently} Hirsch56 _{and Jemai et al.}57 _{used a similar formal scheme to} develop a self-consistent particle-particle RPA.

First, the equations of motion are derived within the stan-dard HFB approximation by computing the time evolution of the density operator44,46 or alternatively from a time-dependent variational principle.45_{In the present case, where} the normal system has small pairing fluctuations, the equa-tions of motion have a simple form:

ıប⳵␬pp⬘ ⳵t = ⳵FT ⳵␬pp⬘ ⴱ , ıប⳵␬hh⬘ ⳵t = ⳵FT ⳵␬hh⬘ ⴱ . 共27兲

In the same spirit as the nuclear physics treatment of effec-tive Hamiltonians that are density dependent, the HFB en-ergy appearing on the right-hand side has been replaced by the GA energy. FT is the total energy functional after the

one-body part has been expanded using Eq. 共23兲 and the

remaining part has been put in the basis that diagonalizes the Gutzwiller Hamiltonian using the transformation of Eq.共21兲.

It is interesting that defining

qi=

冑

ប 2共␬i+␬i ⴱ_兲 and pi= 1 ı

冑

ប 2共␬i−␬i ⴱ_兲,

where i stands for pp

⬘

or hh

⬘

, the equations of motion take the form of classical共Hamilton兲 equations of motion:

⳵qi ⳵t = ⳵FT ⳵pi −⳵pi ⳵t = ⳵FT ⳵qi .

Thus our approximation consists of replacing the generalized forces, computed in the HFB approximation, by forces com-puted in the GA. Previous and present experiences show that, due to the good accuracy of the GA energy and its deriva-tives, the dynamics get substantially improved.

We define the␯th pp-RPA eigenstate of the N + 2–particle system by 兩N + 2,␯典 = R_␯†_{兩N,0典, 共28兲} R_␯†=

兺

p,p⬘ X_pp␯ _⬘a_p†_↑a_p†_⬘↓−

兺

h,h⬘ Y_hh␯_⬘a_h†_⬘↑a_h†_↓, 共29兲 where a_p␴† and a_h␴† create particles and holes in the single-particle levels of Gutzwiller Hamiltonian 共19兲. The states

兩N,␯典 are unprojected states in the sense of Ref.18; i.e., they are auxiliary objects that have particle-particle RPA correla-tions but lack Gutzwiller correlacorrela-tions. This is because they result from creating particle-particle excitations on top of 兩␾0典 and not of 兩⌽G典. In the same way, the␩th eigenstate of

the N − 2–particle system can be represented as

兩N − 2,␩典 = R_␩†_{兩N,0典, 共30兲} R_␩=

兺

hh⬘ X_hh␩_⬘a_h†_↓a_h†_⬘↑−

兺

pp⬘ Y_pp␩_⬘a_p†_⬘↓a_p†_↑, 共31兲 where 兩N,0典 is the unprojected RPA ground state of the N-particle system defined by

R_␯兩N,0典 = R_␩兩N,0典 = 0.

We adopt the convention that ␯ 共␩兲 runs over the np 共nh兲

excitations of the N + 2 共N−2兲–particle system. Within the pp-RPA scheme, X and Y amplitudes can be associated with the following unprojected matrix elements:

X_pp␯ _⬘=具N,0兩a_p⬘↓a_p↑兩N + 2,␯典, Y_hh␯_⬘= −具N,0兩a_h↓ah⬘↑兩N + 2,␯典,

X_hh␩_⬘=具N − 2,␩兩ah↑ah↓兩N,0典, Y_pp␩_⬘= −具N − 2,␩兩a_p↑ap⬘↓兩N,0典.

From the equations of motion for the amplitudes, one can derive the following eigenvalue problem:44,45

冉

关␰p₁↓+␰p₂↑兴␦p₁p₃␦p₂p₄+ Vp₁p₂,p₃p₄ Vp₁p₂,h₃h₄ V_h 1h2,p3p4 ⴱ _−关␰ h₁↓+␰h₂↑兴␦h₁h₃␦h₂h₄+ Vh₁h₂,h₃h₄

冊冉

Wp₃p₄ Zh₃h₄

冊

=

冉

In_p 0 0 − In_h

冊冉

Wp₁p₂ Zh₁h₂

冊

⍀⫾_{, 共32兲}

with In as the n⫻n identity matrix. The matrix elements of

the potential V are obtained from Eq.共24兲 by transforming to

the GA representation with the help of Eq.共21兲.

Equation 共32兲 yields np+ nh eigenvectors which can be

normalized as 兩W兩2_−兩Z兩2_{=⫾1. The sign of the norm allows}

one to distinguish the np addition eigenvectors 共positive

norm兲 from the nhremoval eigenvectors共negative norm兲.45

In the following, we will denote by E_␯共N兲 and E_␩共N兲 the N-particle energies of Hamiltonian 共1兲 when the ␮ term is absent and write the ␮ contribution explicitly, so that the eigenvalues of Eq. 共1兲 are given by E_␩共N兲−␮N. The eigen-values and eigenvectors can be identified with the excitation energies and the amplitudes in the following form:

⍀_␯+_{= E}

␯共N + 2兲 − E0共N兲 − 2␮, W_pp␯_⬘= X_pp␯ _⬘,

Z_hh␯_⬘= Y_hh␯ _⬘, 共33兲

for two-particle addition and

⍀_␩−

= E0共N兲 − E␩共N − 2兲 − 2␮, W_pp␩_⬘= Y_pp␩_⬘,

Z_hh␩_⬘= X_hh␩_⬘, 共34兲

for two-particle removal.

Stability requires that ⍀_␯+⬎0⬎⍀_␩−. Thus if there exists a chemical potential␮such that these two conditions are sat-isfied, the system is stable; otherwise a pairing instability arises. In a finite system, the allowed range of ␮shrinks to zero as an instability is approached. From Eqs.共33兲 and 共34兲,

one sees that stability requires E0共N+2兲−E0共N兲⬎E0共N兲

− E0共N−2兲. This coincides with the stability against a

trans-fer of a pair of particles from one cluster to another one in an ensemble. Note that it does not necessarily coincide with the stability against single-particle transfers. In the thermody-namic limit, both conditions coincide with the well-known stability condition that the compressibility must be positive. We are now in the position to evaluate the pairing corre-lation function within the TDGA. We are interested in on-site s-wave共s兲 pairing and intersite pairing:

⌬i s = ci↓ci↑, 共35兲 ⌬i␦= 1

冑

2共ci+␦↓ci↑+ ci↓ci+␦↑兲, 共36兲 where␦= x , y and i +␦is a shorthand for the first neighbor of site i in the␦ direction.

The dynamical pairing correlations can be computed from

Pij␣␤共␻兲 = − ı Ns

冕

−⬁ ⬁ dteı␻t_具T⌬ i ␣_{共t兲关⌬} j ␤_共0兲兴†_典 = 1 Ns

兺

␯ 具N,0兩⌬¯i␣兩N + 2,␯典具N + 2,␯兩共⌬¯␤j兲†兩N,0典 ␻−⍀_␯++ ı0+ − 1 Ns

兺

_␩ 具N,0兩共⌬¯␤j兲†兩N − 2,␩典具N − 2,␩兩⌬¯i␣兩N,0典 ␻−⍀_␩−− ı0+ , 共37兲 where Ns denotes the number of sites of the system and ␣

= s , x , y. The average under the integral is done on the Gutzwiller projected RPA state. The latter is our best esti-mate for the true ground state of the system. The matrix elements in the rightmost term are done on the unprojected states. In the spirit of the GA, Gutzwiller projections are effectively taken into account by a renormalization of the operators. As usual, local operators do not get renormalized, i.e.,⌬¯i

s_⬅⌬ i s

, whereas nonlocal operators acquire a renormal-ization through the z factors:

⌬¯i␦⬅

1

冑

2共zi+␦↓zi↑ci+␦↓ci↑+ zi↓zi+␦↑ci↓ci+␦↑兲. 共38兲 This is similar to the renormalization of the GA current op-erator in the computation of the optical conductivity.18 _The validity of this prescription can be checked using sum rules as discussed below. Obviously in the HF theory based on BLA, bare operators are used in Eq.共37兲.

The matrix elements of the pairing operators can be ob-tained from the amplitudes X and Y by using transformation 共21兲. In Eq. 共37兲 the first and second terms represent the

correlations in case of two-particle addition and removal, respectively.

In the case of a separable potential Vk₁k₂,k3k4=vk1k2vk3k4

ⴱ _,

the pp-RPA equations can be easily solved. In particular for the case of a paramagnetic solution analyzed in Ref. 41, the TDGA interaction kernel关Eq. 共24兲兴 involves only local pairs,

␦Fpp= V

兺

i ␦具ci†_↑ci†_↓典␦具ci↓ci↑典, 共39兲 with V =U − 2⌺␴ 1 − n , 共40兲

where n⬅N/Nsand⌺␴is defined in Eq.共20兲. In this case the

on-site response can be found in momentum space,

Pii␣␣共␻兲 = 1 Ns

兺

q P_␣␣共q,␻兲, P_␣␣共q,␻兲 = − ı Ns

冕

−⬁ ⬁ dteı␻t具T⌬q␣共t兲关⌬q␣共0兲兴†典, 共41兲 with共␣= s兲,

⌬q␣= 1 Ns

兺

j

⌬i␣eiq.Rj, 共42兲

and is given by an effective ladder-type equation:

Pss共q,␻兲 = Pss0共q,␻兲 1 − VPss0共q,␻兲 , 共43兲 Pss 0_{共q,␻兲 =} 1 Ns

兺

k 1 − f共␰k兲 − f共␰k+q兲 ␻−␰k−␰k+q+ ı␩k,k+q , 共44兲

with f共␰k兲 as the Fermi distribution function and ␩k,k⬘ ⬅0+_sgn共␰

k+␰k⬘兲. The single-particle energies are given by ␰k=␧k−␮, where␧k⬅z02ek+⌺Gis the GA dispersion relation

and ek=兺jtije−ık.共Rj−Ri兲 is the bare one and z0is the hopping

renormalization factor 关Eq. 共15兲兴 evaluated at the saddle

point.

We are interested also in the fluctuation response for in-tersite bond singlets with s-wave symmetry 共␣= +兲 or d-wave 共␣= −兲 symmetry: ⌬i␣= 1 2共⌬i x +⌬i −x +␣⌬i y +␣⌬i −y_兲.

In this case the ladder series takes the following matrix form: Pគគ 共q,␻兲 = Pគគ0共q,␻兲 + Pគគ0共q,␻兲⌫គគPគគ共q,␻兲, 共45兲 with Pគគ 共q,␻兲 =

冉

Pss共q,␻兲 Ps␣共q,␻兲 P_␣s共q,␻兲 P_␣␣共q,␻兲

冊

, ⌫គគ =

冉

V 0 0 0

冊

. Thus P_␣␣共q,␻兲 = P_␣␣0 共q,␻兲 +P␣s 0 _{共q,␻兲VP} s␣ 0_共q,␻兲 1 − VPss0共q,␻兲 , 共46兲 and the poles of P_␣␣共q,␻兲 are determined by the local pair correlation 关1−VPss

0_{共q,␻兲=0兴 and the bare correlation.}

Equations共43兲–共46兲 are also valid in the BLA approach,

re-placing the Gutzwiller local self-energy with the HF one 共⌺_␴= Un/2, z₀= 1兲 and taking V=U.

The exact on-site s-wave spectral density satisfies the fol-lowing sum rules:

− 1 ␲

冕

0 ⬁ d␻Im Pii ss_{共␻兲 = 1 − n + 具n} i_↑ni_↓典, 共47兲 − 1 ␲

冕

−⬁ 0 d␻Im Pii ss_{共␻兲 = − 具n} i_↑ni_↓典. 共48兲

In our case these sum rules allow us to compute RPA correc-tions to the double occupancy, and they will be used below to evaluate RPA corrections to the GA ground-state energy. Note that the total spectral weight is equal to 1 − n.

In addition the following energy-weighted sum rule is sat-isfied: M₁ss共⬁兲 = 2 Ns 具T典GA+共2␮− U兲共1 − n兲, 共49兲 M₁ss共␻兲 = 1 ␲Ns

兺

ij

冕

−⬁ ␻ d␻¯␻¯ Im Pij ss_共␻ ¯兲, 共50兲

with T as the kinetic energy. This sum rule is satisfied exactly within the present pp-RPA in the sense that the right- and the left-hand sides are equal provided that the kinetic expecta-tion value on the right-hand side is computed at the GA level. Thus in contrast with Eqs.共47兲 and 共48兲, this sum rule

provides no new information but is useful in performing a consistency check. The same prescription is valid in the con-ventional HF plus pp-RPA approach, where the right-hand-side expectation values have to be computed at HF level.58 As shown in Appendix C, the consistent renormalization of intersite operators can be checked from an analogous sum rule.

III. RESULTS

In this section we present results for pair correlations in the repulsive Hubbard model within the framework devel-oped above. Since the RPA-type scheme used in the TDGA differs in some regard from the approaches usually invoked in solid-state theory, we first illustrate the method for a two-site model which also can be solved exactly. Due to the mean-field character of the present approximations, the method is expected to improve with dimensionality so this zero-dimensional example represents the worst case and al-lows us to give a first check of the potentialities and limita-tions of the method. Finally, we compare our method with exact results on small clusters and demonstrate its superior performance as compared to the BLA.

A. Two-site model

In order to illustrate the formalism, we consider a two-site model with two particles having up- and down-spins. The ground-state wave function reads as

兩2,0典 =␣兩S典 +␤兩D典, 共51兲 where 兩S典 =

_冑

1 2共兩1↑2↓典 − 兩1↓2↑典兲, 共52兲 兩D典 =

_冑

1 2共兩1↑1↓典 + 兩2↑2↓典兲, 共53兲 and the corresponding amplitudes and the ground-state en-ergy E0共2兲 are given by

␣=

_冑

2t ␻02+ 4t2 , 共54兲 ␤=

_冑

−␻0 ␻02+ 4t2 , 共55兲

E₀共2兲 =␻₀⬅U

2

冉

1 −

冑

1 + 16t2

U2

冊

. 共56兲

For four and zero particles, there is only one state with en-ergy E共4兲=2U and E共0兲=0, respectively. The enen-ergy differ-ences between two-particle addition共removal兲 states and the ground state are

⍀+_{= E共2 + 2兲 − E} 0共2兲 − 2␮= U 2 + 1 2

冑

U 2_{+ 16t}2_, ⍀−_{= E} 0共2兲 − E共2 − 2兲 − 2␮= −⍀+,

and the chemical potential is taken as␮= U/2, which is the exact value at half filling at an infinitesimal temperature. The corresponding matrix elements for local 共s兲 and intersite 共x兲 pairing operators ⌬s= 1

冑

2共c1↓c1↑+ c2↓c2↑兲, 共57兲 ⌬x= 1

冑

2共c1↓c2↑− c1↑c2↓兲 共58兲 read as 具2,0兩⌬s兩2 + 2典 = 具2,0兩⌬s †_{兩2 − 2典 =␤} , 共59兲 具2,0兩⌬x兩2 + 2典 = 具2,0兩⌬x†兩2 − 2典 =␣. 共60兲

Note that there is only one state with 2⫾2 particles so we dropped the excitation index. One can check that the sum rules in Eqs.共47兲–共49兲 are satisfied. For example, the double

occupancy is given by␤2/2 and the first moment sum rule − 2 Ns 具T典 − 共2␮− U兲共1 − n兲 =⍀+_{兩具2,0兩⌬} s兩2 + 2典兩2−⍀−兩具2,0兩⌬s†兩2 − 2典兩2 = 2共U −␻0兲␤2= −具T典

is also fulfilled, as it should be.

Consider now the same model within the TDGA. On the GA level, one finds two single-particle states for each spin direction which at half filling can be put as

␰h␴= − t共1 − u2兲 + ⌺␴−␮, 共61兲 ␰p␴= t共1 − u2兲 + ⌺␴−␮, 共62兲

where the h state is occupied with a spin-up and a spin-down particle and we have defined u⬅U/UBR, with UBR= 8t. For U⬍U_BR there is a paramagnetic solution which becomes insulating at the Brinkmann-Rice transition point UBR.59

For u⬎

冑

2 − 1 the more stable solution is an antiferromag-netic broken-symmetry solution which does not have a Brinkmann-Rice transition point.

For the paramagnetic solution, one has spin-independent Lagrange multipliers which from the Kotliar-Ruckenstein共or Gebhard’s兲 scheme are obtained as

⌺␴⬅ ⌺0=␮= U

2. 共63兲

Since there is only one two-particle addition and removal state, we have to diagonalize the following 2⫻2 pair fluc-tuation RPA problem:

冉

2t共1 − u2_{兲 + V/2 V/2} V/2 2t共1 − u2_{兲 + V/2}

冊冉

X Y

冊

=⍀

冉

1 0 0 − 1

冊冉

X Y

冊

. 共64兲 The interaction V corresponds to the local part in the expan-sion in Eq. 共24兲 共note that the first derivatives of Ai0vanish

for a paramagnetic solution兲 as follows: V = − 2t共1 − u2兲A

⬙

+ U

1 − n= 4tu共2 − u兲 1 + u

1 − u, 共65兲 where A

⬙

is defined in Appendix B. Here, the diverging part, proportional to 1/共1−n兲, is canceled by an analogous contri-bution in the first term. The same result can be obtained from Eq. 共40兲 by taking the limit n→1,41 _{and it coincides in} modulus with the effective interaction at half filling in the particle-hole case.16–18 _{This is consistent with the fact that} the attractive-repulsive transformation converts particle-particle fluctuations in particle-particle-hole fluctuations with equal interaction but sign reversed.60

Diagonalization of Eq. 共64兲 yields the two eigenvalues

⍀⫾_{= ⫾ 2t共1 − u}2_兲

冑

_{1 +} V

2t共1 − u2_兲. 共66兲

For the eigenvectors, the following relations hold:

共X⫾_{+ Y}⫾_兲2_{= ⫾}2t共1 − u 2_兲

⍀⫾ , 共67兲

共X⫾_兲2_−共Y⫾_兲2_{= ⫾ 1, 共68兲}

from which one can compute the amplitudes for the local and intersite pairing correlations between unprojected states,

具2,0兩⌬s兩2 ⫾ 2典 = 1

冑

2共X ⫾_{+ Y}⫾_{兲, 共69兲} 具2,0兩⌬x兩2 ⫾ 2典 = − 1

冑

2共X ⫾_{− Y}⫾_{兲. 共70兲}

Finally, the expectation values between Gutzwiller projected states are the same as for unprojected states in the case of⌬s

and should be renormalized in the intersite case, i.e., ⌬¯x

=共1−u2_兲⌬

x. See Appendix C for a consistency check of this

prescription.

For the local pairing operator, we find that first moment sum rule 共49兲,

⍀+_{兩具2,0兩⌬} s兩2 + 2典兩2−⍀−兩具2,0兩⌬s †_{兩2 − 2典兩}2 = 2t共1 − u2_{兲 = −} 2 N具T典GA− 2␮共1 − n兲 + U共1 − n兲, 共71兲 is satisfied共note that n=1兲 in the TDGA, as anticipated.

The TDGA two-particle addition and removal energies are displayed in Fig.1and compared with the exact ones. Solu-tions obtained in the paramagnetic regime are shown as solid lines. Remarkably, due to a cancellation of共1−u兲 terms, the Brinkmann-Rice transition does not reflect in the excitation energies of the paramagnetic phase, which become soft at a much larger value of the repulsion 共u=1+

冑

2兲. Instead, the transition shows up in the TDGA pairing correlations共right panel of Fig.1兲. Indeed, the Gutzwiller renormalization

fac-tors zi␴ drive the matrix elements of the intersite pairing

operator to zero at the Brinkmann-Rice transition point. It is interesting that neglecting zi_␴in the pairing operator, the

sup-pression is replaced by a divergence so that the zi␴cancel the

unphysical divergence but overcorrect it. This problem is partially cured in the broken-symmetry state 共dashed lines兲. However, for U→⬁ the matrix element tends to 1/2, whereas the exact result approaches 兩具2−2兩⌬x兩2,0典兩2→1

since the ground state can be written as⌬x

†_{兩2−2典 in this limit.}

Instead, the ground state with broken symmetry has only one configuration of the two that generate the singlet ground state, which explains the 1/2 factor. For the local pairing operator, the behavior is better. We notice that the TDGA excitations energies are in quite good agreement with the exact values, especially when one allows for the broken-symmetry solutions.

Finally we can use Eq.共47兲 to compute the pair

fluctua-tion derived double occupancy:

DRPA_=具n

i↑ni↓典 = 具2,0兩ci↓ci↑兩2 + 2典具2 + 2兩ci†_↑ci†_↓兩2,0典.

共72兲 Figure2共left panel兲 shows the double occupancy in different

approximations. HF completely neglects correlations; hence the double occupancy is independent of U. The BLA based correction drives the approximation much closer to the exact result at small U but strongly underestimates the correction at large U. In the GA, correlations are already taken into account at the static level and the double occupancy is strongly suppressed as a function of U. Thus the pp-TDGA correction is small and brings the double occupancy much closer to the exact one than its HF based counterpart, in a much larger range of interaction. The fact that the pp-TDGA is close to the GA double occupancy indicates that the theory is nearly self-consistent. In Ref. 41 it was shown that this feature is enhanced in two dimensions, pointing to an im-provement of the performance as the dimensionality is in-creased. In contrast the BLA is clearly quite far from being self-consistent in this sense.

For large U, the exact double occupancy is of order UJ ⬃t2_/U2 _{due to the same charge fluctuations that build the}

double exchange interaction J.61 _{Since the GA and the} TDGA results in Fig. 2 are for paramagnetic solutions, the double occupancy vanishes at the Brinkman-Rice point. This makes clear that the scale J is not present in the paramag-netic GA or TDGA.

The above results allow us to compute the pp-RPA ground-state energy by using the coupling constant integra-tion trick,62,63

E₀RPA= − 2t + UNs

冕

0

U

dU

⬘

DRPA共U

⬘

兲. 共73兲 Here the first term is the ground-state energy for U = 0. Re-stricting on the paramagnetic solutions, we find for the TDGA and BLA

E₀TDGA= − 2t + 4t共

冑

1 + 2u − u2− 1兲, 共74兲 0 0.25 0.5 0.75 1 u=U/(8t) -10 -5 0 5 10 Ω ± /t exact TDGA (para) TDGA (AF) 0 0.25 0.5 0.75 1 u=U/(8t) 0 0.2 0.4 0.6 0.8 1 |<2| ∆ |2 ±2 >| 2

FIG. 1. 共Color online兲 Left panel: Excitation energies for two-particle addition and removal computed for the two-site model with two particles. Right panel: Matrix element of the local 共lower curves兲 and intersite 共upper curves兲 pairing operators. The underly-ing GA saddle points are paramagnetic共p兲 共dashed-dotted lines兲 and antiferromagnetic共dashed lines兲. The critical value u=

冑

2 − 1 of the corresponding transition is indicated by the vertical dotted line.

0 0.25 0.5 0.75 1 U/(8t) -2 -1.5 -1 -0.5 0 0.5 1 E0 0 0.25 0.5 0.75 1 U/(8t) 0 0.1 0.2 dou bl e occupancy exact HF BLA TDGA GA

FIG. 2. 共Color online兲 Double occupancy 共left panel兲 and ground-state energy 共right panel兲 of the two-site Hubbard model computed within HF, BLA, TDGA, and exact diagonalization. Only paramagnetic ground states are considered.

E₀BLA= − 2t + 2t共

冑

1 + U/2 − 1兲, 共75兲 which both yield the exact result ␻₀ up to second order in U/t.

Due to the more accurate estimate for the double occu-pancy 关see Fig. 2共a兲兴, it turns out that the TDGA gives a significantly better approximation for the ground-state en-ergy关see Fig.2共b兲兴 than the BLA does over the whole range

of U, before the onset of the Brinkmann-Rice transition. This should be compared with an analogous calculation in the particle-hole channel in Ref. 18. In the latter case, one ob-tains a singularity in E0at u = 8共

冑

2 − 1兲 due to the onset of the

antiferromagnetic ground state. Since there is no instability in the particle-particle channel, such problems do not arise in the present case. Thus it seems convenient in general to com-pute RPA corrections to the energy in a channel free from instabilities.

B. Comparison with exact diagonalization

In this section, we study the dynamical pairing correla-tions within the TDGA on small clusters and compare our results with the BLA approach and exact diagonalizations. We start by computing the long-distance pairing correlations 具⌬i␦⬘共⌬j␦兲†典 for bond singlets. Within the pp-RPA these

corre-lations are obtained by integrating the addition part of the dynamical spectral function:

具⌬i␦⬘共⌬j␦兲†典 =

1 ␲

冕

0

⬁

d␻Im Pji␦⬘␦共␻兲.

Thus the comparison with the exact diagonalization results provides a stringent test of the total spectral weight. We re-mind the reader that in the TDGA the singlet operator is renormalized by the zi␴factors.

In Fig.3we show results for 10 particles on a cluster with 18 sites, tilted by 45°, and, therefore, having all the spatial symmetries of the infinite lattice. A particular representation of this cluster, with its boundary conditions, is shown in the upper panel of Fig. 3. In this case, the GA ground state is paramagnetic. We concentrate on nonoverlapping singlets and the distance between them is measured from their cen-ters.

The two singlet operators can form a perpendicular con-figuration, such as “s-b” in the upper panel of Fig. 3, or a parallel configuration, such as “s-a” in the same figure. Note that for Rij= Ri− Rj= 2, there are two points in the lower

panel corresponding to two parallel configurations in which one of the singlets is displaced either along the x or along the y direction 共labeled “a” in the upper panel of Fig.3兲.

In Fig. 3, the vertical bars point to the GA value of the singlet correlations, i.e., the decoupled result of具⌬i␦⬘共⌬j␦兲†典GA

but still renormalized with the zi␴ factors. In the same spirit

of Ref.25, the length and orientation of the bars reflects the “vertex contribution.” This quantity measures the correlation induced interaction between two singlets, which is attractive when the TDGA value is larger than the one computed within the bare GA. For nearby singlets we observe excellent agreement between the exact diagonalization result and the

TDGA for both U/t=4 and 10 共see lower panels of Fig.3兲.

In this case we also observe an effective attractive interac-tion. In general, with increasing distance TDGA overesti-mates the exact correlations, and the difference becomes more pronounced for larger U/t. This behavior can be ex-pected due to the fact that the Gutzwiller method is based on a local projector which neglects intersite correlations. One can therefore anticipate that the incorporation of intersite projections共as in Jastrow-type wave functions64,65_{兲 into the} TDGA would lead to an improvement of the long-distance pair correlations. Nevertheless, it turns out that the TDGA yields a rather good description of long-distance pair corre-lations共especially for moderate values of U/t兲 as compared to exact diagonalizations.

In order to compare dynamical properties of the TDGA with BLA and exact results, we compute the d-wave and s-wave correlations which are defined in Sec. II B. The com-parison of the results in different approximations can be done by aligning the chemical potentials 共as in the last figure of Ref. 41兲 or aligning the absolute energies. In the following,

we adopt the last procedure by eliminating the chemical po-tential from the response function. We define␻

⬘

=␻+ 2␮so that the poles in P_ii␣␣共␻

⬘兲 occur at

␻

⬘

= E_␯共N+2兲−E₀共N兲 and ␻

⬘

= E_␩共N−2兲−E0共N兲. a a b s 0.5 1 1.5 2 2.5 3 Distance R_ij -0.025 0 0.025 0.05 <∆i ∆ + >j exact (parallel) TDGA (parallel) exact (perp.) TDGA (perp.) 0.5 1 1.5 2 2.5 3 3.5 Distance R_ij U/t=4 U/t=10 (b) (a)

FIG. 3. 共Color online兲 Top panel: Sketch of the 18-site cluster and corresponding boundary conditions. Bottom panels: Bond sin-glet pairing correlations on the 18-site lattice for U/t=4 and 10. The bars at the TDGA symbols point to the bare GA value of the singlet correlations. The separation between bond singlets is defined as the shortest distance between their centers. Taking the singlet “s” as reference, it should be noted that there exist two parallel singlet configurations “a” with distance R_s− R_a= 2. The singlets “b” and “s” form a perpendicular configuration.

In Fig.4, we show the addition spectra for Pii␣␣共␻

⬘

兲

evalu-ated for a 18-site cluster with 10 particles and U/t=10. This corresponds to a closed-shell configuration where the HF and GA approaches yield paramagnetic solutions. The on-site s-wave case Pii

ss_共␻

_⬘

_{兲 has already been analyzed}41 _{for a}

smaller cluster and it is shown in Fig. 4 for 18 sites as a reference since, as explained in Sec. II B, it also determines the overall features of the intersite pairing correlations. Fig-ure5 共main panel, dashed-dotted line兲 displays the effective

interaction V for the present case as a function of U/t. In Ref.41we showed that the on-site pair excitations for large U in both BLA and TDGA are dominated by an anti-bound state whose lower band edge is at ␻

⬘

= U. The band states at energies 1t⬃5t have a very small spectral weight. In the top panel of Fig.4, the solid black arrow indicates the value of 2␮ in the exact case with ␮=共␮++␮−兲/2 and ␮+ = E共N+1兲−E共N兲,␮−_{= E共N兲−E共N−1兲. The position of 2␮in}

the GA 共red dashed arrow兲 is very close to the exact one, whereas in HF 共blue dotted arrow兲 it is shifted to higher energies. Thus, aligning the chemical potentials, the position of the BLA 共TDGA兲 antibound state is in disagreement 共agreement兲 with the exact result.41 _{In addition, the TDGA} gives a good account of the low-energy spectral weight共see inset兲, which is much larger and shifted to higher energies in the BLA. This was understood as due to the different way the self-energy is renormalized by interactions in GA and in HF.41 _{This dramatic difference in performance gets greatly} amplified for intersite correlations 关see Figs. 4共b兲 and4共c兲兴 because the band states acquire significant weight.

The intersite correlations have also significant weight at the energy of the s-wave antibound state␻

⬘

⬃U. This can be understood from the second term in Eq.共46兲; it corresponds

to processes in which an intersite pair decays into an on-site pair. Note that the d-wave pair cannot mix with a local s-wave pair at q = 0 but they couple at finite q.

In addition, the exact result shows a satellite at ␻

⬘

⬇2U for the intersite cases corresponding to a process in which the two particles are created at neighboring sites that are initially

0 5 10 15 20 25 30 ω’/t 0 0.05 0.1 0.15 0.2 0.25 -1/ π Im P( ω ) exact TDGA BLA 0 5 10 ω’/t 0 0.01 0.02 -1/ π Im ∫P( ω ) 0 5 10 15 20 25 30 ω’/t 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 -1/ π Im P( ω ) exact TDGA BLA 0 10 ω’/t 0 0.2 0.4 0.6 -1/ π Im ∫P( ω ) 0 5 10 15 20 25 30 ω’/t 0 0.05 0.1 0.15 0.2 0.25 0.3 -1/ π Im P( ω ) exact TDGA BLA 0 5 10 ω’/t 0 0.05 0.1 -1/ π Im P( ω ) (a) (b) (c)

FIG. 4. 共Color online兲 Imaginary part of the 共two-particle addi-tion兲 pairing correlation function in Eq. 共37兲 for on-site pairing 共a兲 and extended s-wave共b兲 and d-wave 共c兲 symmetries. Results are for 10 particles on an 18-site cluster and U/t=10 for which the under-lying mean-field solution in BLA and TDGA is paramagnetic. The deltalike excitations are convoluted by Lorentzians with width ␧ = 0.2t. The arrows in the upper panel point to the chemical poten-tials obtained within the various methods. The insets detail the fre-quency evolution of the weight at small energies.

0 5 10 15 20 U/t 0 10 20 30 40 50 Vii,ii [t] 0 10 20 30 -0.03 -0.02 -0.01 0 V_ii,jj[t] 0 5 10 15 20 -0.4 -0.20 0.2 0.4 0.6 V_ii,ij[t] N=10 param. N=18 SDW N=18 param.

FIG. 5. 共Color online兲 Main panel: Local part of the effective interaction Vii,ii关see Eq. 共24兲兴 for the 18-site cluster. The solid and

dashed lines refer to the half-filled cluster with paramagnetic and spin-density wave ground state, respectively. The dashed-dotted line is for the ten-particle system with a paramagnetic ground state. Upper left inset: Effective interaction between pairs on nearest-neighbor sites V_ii,jj. Lower right inset: Effective interaction V_ii,ij between a local pair on site Riand an intersite pair on the bond

defined by R_iand R_j. V_ii,ij⬎共⬍兲0 for S_iz⬍共⬎兲0 共solid and dashed lines, respectively兲.

single occupied, thus leading to two on-site pairs in the final state. This can be visualized as the creation of the two par-ticles in the upper Hubbard band. Since this band is not present in our starting point共i.e., GA兲, this satellite is absent in the pp-RPA. This feature reappears if one starts from a GA state that has both lower and upper Hubbard bands at the cost of spontaneous symmetry breaking. This case is analyzed in the following.

We now consider the half-filled system where both HF and GA have a spin-density wave共SDW兲 ground state. The dynamical pairing correlations, for 18 particles in the 18-site cluster, are shown in Fig.6. Again, we are comparing abso-lute energies, but the position of 2␮in the different approxi-mations is aligned because the chemical potentials coincide and are equal to the exact result␮= U/2.

For local s-wave, extended s-wave, and d-wave fluctua-tions, the exact result has a broad distribution of weight cen-tered around ␻

⬘

⬃2U. Clearly this corresponds to the high-energy satellite of the previous case. Now practically all sites are singly occupied so the probability to find an empty site where to create a local s-wave pair is very small and there is practically no weight at the energy of the antibound state ␻

⬘⬃U. Indeed, if the TDGA computation is done with a

paramagnetic state 共which cannot reproduce the satellite at 2U兲, the response becomes identically zero.41 _{This can be} also seen from the sum rule in Eqs.共47兲 and 共48兲 since in the

paramagnetic TDGA and GA, the double occupancy van-ishes for the half-filled system above the Brinkman-Rice transition point.

Breaking the symmetry and allowing for the SDW, both BLA and TDGA give a good estimate of the energy scale but with a more narrow distribution of weight. Note that the on-site s-wave response has a much smaller intensity, con-sistent with the fact that the feature originates from intersite excitations which get mixed with the on-site response.

Processes in which the two particles are created on differ-ent singly occupied sites are allowed in both approximate theories, and they lead to approximately similar results. In this case the interaction is practically ineffective. In fact, the excitation spectra evaluated from P0 _{共from the bare}

Gutzwiller Hamiltonian兲 shown as thin lines in Fig.6differ only slightly from the TDGA result. The latter shows only a small shift of spectral weight to higher energies due to the inclusion of particle-particle correlations. This is in contrast to the paramagnetic case away from half filling共see Fig.4兲,

where all high-energy共antibound兲 states are determined from poles in P共␻

⬘

兲 but are absent in the noninteracting case P0共␻

⬘

兲 as discussed before.

The insets of Fig.6show the evolution of the two-particle spectral weight. For both extended s-wave and d-wave sym-metries, the total weight approaches 1/2 in the pp-RPA theo-ries in the limit U→⬁ as for the AF solution of the two-site model in the right panel of Fig.1. This can be easily under-stood from the Néel limit. In contrast, the exact result has a larger weight, which can be understood from quantum fluc-tuations that induce spin flips closely related to the reduction of the magnetization in a quantum antiferromagnet. In the extreme case of the two-site model, the fluctuations lead to a local singlet and the exact spectral weight for U→⬁ is twice the approximate one as discussed in connection with Fig.1.

On the contrary, increasing the dimensionality, the quantum fluctuations are suppressed quickly, driving the exact result to the TDGA result.

The structure of the TDGA interaction kernel关Eq. 共24兲兴 is

more complex than the paramagnetic case. Figure5displays

10 20 30 ω’/t 0 0.005 0.01 0.015 0.02 -1/ π Im P( ω ) exact TDGA BLA GA 10 15 20 25 30 ω’/t 0 0.02 0.04 0.06 -1/ π Im ∫P( ω ) 10 15 20 25 30 35 ω’/t 0 0.05 0.1 0.15 0.2 0.25 -1/ π Im P( ω ) exact TDGA BLA GA 10 15 20 25 30 ω’/t 0 0.2 0.4 0.6 -1/ π Im ∫P( ω ) (a) (b) 10 15 20 25 30 ω’/t 0 0.05 0.1 0.15 0.2 0.25 0.3 -1/ π Im P( ω ) exact TDGA BLA GA 10 15 20 25 30 ω’/t 0 0.2 0.4 0.6 -1 /π Im ∫P( ω ) (c)

FIG. 6. 共Color online兲 Imaginary part of the 共two-particle addi-tion兲 pairing correlation function 关Eq. 共37兲兴 for on-site pairing 共a兲 and extended s-wave共b兲 and d-wave 共c兲 symmetries. Results are for a half-filled 18-site cluster and U/t=10. The underlying mean-field solution in BLA and TDGA is a spin-density wave. The deltalike excitations are convoluted by Lorentzians with width␧=0.5t. The insets detail the frequency evolution of the two-particle spectral weight.

the corresponding nonvanishing elements as a function of U/t. For small on-site repulsion, the local contribution be-haves as Vii,ii⬇U, independent of the filling and the ground

state, as it should be. With the onset of the SDW at U/t ⬇4.1, Vii,ii 共dashed line兲 starts to deviate from the

corre-sponding interaction in the paramagnetic limit 共solid line兲, which diverges at the Brinkmann-Rice transition.41 Interest-ingly, the interaction between local pairs on adjacent sites 共upper left inset of Fig. 5兲 is always attractive in the SDW

phase with a maximum attraction at U/t⬇10. In addition one finds an attractive or repulsive interaction between a lo-cal pair on site i and an intersite pair ci,↑cj,↓. For i and j

nearest neighbors, the interaction is attractive 共repulsive兲 if the pair has the same 共opposite兲 spin with respect to the underlying Néel magnetic moments of sites i and j. Never-theless, these additional fluctuations in the TDGA cannot overcome the strong residual on-site repulsion so that the system remains stable against a transition toward supercon-ductivity. This also holds for a SDW ground state away from half filling.

Finally, as in the two-site example, we calculate the en-ergy correction from the TDGA for a half-filled 4⫻4 cluster from Eqs. 共47兲 and 共73兲. In Fig. 7, we compare the corre-sponding result for the particle-particle and particle-hole channel 共from Ref. 17兲 with the bare GA and the exact

ground-state energy. The underlying saddle point of the GA solution is a SDW. As can be seen from Fig.7, the particle-hole TDGA correction is approximately twice that in the particle-particle channel. The former gives a quite accurate approximation for intermediate values of the on-site repul-sion but tends to overshoot the exact result for large U/t 共note that these energy corrections are not derived from a variational principle and thus do not constitute an upper bound for the exact result兲. With the considered range of U/t, the particle-particle corrections always are slightly higher in energy than the exact ground state. However, in comparison with the HF+ RPA energy corrections, which are

by far too large,17_{the TDGA yields a reasonable} approxima-tion to E0in both particle-hole and particle-particle channels.

IV. CONCLUSIONS

In this paper, we have extended the TDGA toward the inclusion of pair correlations in the Hubbard model. The present analysis is complementary to previous computations in the particle-hole channel17–19_{where we have analyzed the} spectrum of charge and spin excitations. In comparison with exact diagonalization results, the TDGA yields very good agreement for the dynamical pair-correlation function, espe-cially for the low-energy excitations and away from half fill-ing, where it performs significantly better than the HF based BLA theory.

Compared to numerical methods,66 _{our approach can be} pushed to much larger systems. In particular, it is suitable for the evaluation of pair correlations in the negative-U Hubbard model, where the approach is at least qualitatively capable of capturing the crossover from weak-coupling BCS to strong-coupling behavior.67_{An important outcome of this} investiga-tion concerns the justificainvestiga-tion of the antiadiabatic assumpinvestiga-tion for the time evolution of the double occupancy, which was needed in the charge particle-hole channel.

For the Hubbard Hamiltonian with particle-hole symme-try共e.g., nearest-neighbor hopping at half filling兲, the super-conducting instability and the charge-density wave instability in the particle-hole channel are degenerate. Evaluation of the latter instability within the TDGA requires invoking the an-tiadiabaticity condition, whereas the expansion in the particle-particle channel of Eq.共24兲 goes without it. The fact

that the two calculations within the TDGA for the particle-hole and the particle-particle sectors give the correct degen-eracy of the instabilities for a charge-rotational invariant sys-tem clearly indicates that the antiadiabaticity assumption was indeed correct, at least in the limit of small frequencies. That is, other possibilities, as keeping the double occupancy fixed at the stationary value共rather than following the time evolu-tion of the density matrix兲, would have led to an unphysical breaking of charge-rotational symmetry. This is consistent with Vollhardt,16_{who derived the static susceptibilities of the} paramagnet by using the same assumption.

Our approach does not produce a superconducting insta-bility in the Hubbard model. This is because, in the paramag-netic phase, the effective interaction does not contain the attractive part due to the exchange of spin fluctuations. In-deed, the superexchange scale J is absent in the paramag-netic GA. On the other hand, it turned out that the effective interaction in the SDW phase contains attractive contribu-tions between nearest-neighbor pairs which, however, are too weak to yield a superconducting instability once the SDW is present.

The flexibility of the present approach allows one to study collective modes in inhomogeneous superconducting states once the Hamiltonian is augmented with a suitable pairing potential. This may find application in the physics of high-Tc

cuprates, where in many compounds the occurrence of elec-tronic inhomogeneities 共e.g., in the form of stripes兲 is now well established. In addition, the method opens the

possibil-0 5 10 15 20 U/t -1.5 -1 -0.5 0 E/N particle-hole particle-particle GA exact

FIG. 7. 共Color online兲 Ground-state energy of the half-filled 4 ⫻4 Hubbard model computed within GA, TDGA in the particle-hole channel, TDGA in the particle-particle channel, and exact diagonalization.