Projection Operator Method

Nikolay M. Plakida

Abstract A general projection operator method in the equation of motion method for the two-time Green functions is formulated. An exact Dyson equation for an arbitrary Green function is derived. The method is used to consider the single-particle electron Green functions for the Hubbard model within the non-crossing approximation for the self-energy. Strong-coupling superconductivity theory for the model is formulated and an equation for the superconducting Tc is analyzed. We argue that the d-wave pairing with high-Tc can occur in the repulsive Hubbard model, which is mediated by the antiferromagnetic exchange interaction and the spin-fluctuation scattering, both induced by the kinematic interaction for the Hubbard operators. A microscopic theory for the dynamic spin susceptibility within the projection operator method is formulated. The theory is applied to study spin-excitation spectrum for the t-J model within the mode-coupling approximation for the self-energy. A new approach to the theory of the magnetic resonance mode in cuprate superconductors is proposed.

6.1 Introduction

One of the basic models for studies of electronic spectra and superconductivity in strongly correlated electronic systems (SCES), as the cuprate high-temperature superconductors, is the Hubbard model [1]. In the simplest approximation, the model is specified by two parameters: a single-electron hopping matrix element t between the nearest neighbors and a single-site Coulomb energy U:

Joint Institute for Nuclear Researech, Dubna, Moscow region 141980, Russia e-mail: [email protected]

A. Avellaand F. Mancini (eds.), Strongly Correlated Systems, Springer Series in Solid-State Sciences 171, DOI 10.1007/978-3-642-21831-6_6, © Springer-Verlag Berlin Heidelberg 2012

where a}a(aia) are the creation (annihilation) operators for an electron of spin a on the lattice site i and nia = ajaaia is the electron occupation number. The model (6.1) permits one to consider both cases of weak correlations, U ^ t, and of strong correlations, U t. In the weak correlation limit, a metallic state is observed, while in the strong correlation limit the model describes a Mott-Hubbard insulating state at half-filling (an average occupation electron number n = 1). For hole doping (n < 1) of the lower Hubbard subband (LHB), or for electron doping (n > 1) of the upper Hubbard subband (UHB) the model describes a strongly correlated metal.

Various methods have been used to investigate the Hubbard model, among which are numerical simulations for finite clusters (see, e.g., [2,3]), dynamical mean field theory (DMFT) (see, e.g., [4, 5]), the dynamical cluster theory (see, e.g., [6,7]), and other methods discussed in other chapters of this book. A rigorous analytical method in the limit of strong correlations is based on the Hubbard operator (HO) technique [8] since in this representation the local constraint of no double occupancy of any lattice site is rigorously implemented by the Hubbard operator algebra. However, since the HOs are composite operators with complicated commutation relations, the diagram technique for them is rather complex and only simplest set of diagrams can be taken into account in calculations (see, e.g., [9]).

A more convenient and straightforward technique is based on the equation of motion method for the thermodynamic two-time Green functions (GFs) introduced by Bogoliubov and Tyablikov [10]. By sequential differentiating GFs over time t or t', a chain of equations can be derived. To obtain a closed system of equations, an approximation should be used for higher order GFs, usually called as "truncating" or "decoupling" of GFs [11,12]. We emphasize here that any type of a decoupling of higher order GFs for Fermi or Bose operators corresponds to a certain set of diagrams in the temperature diagram technique for the causal GFs (see, e.g., [13]). This enables to evaluate which set of diagrams are taken into account in the decoupling approximation and, therefore, to estimate the accuracy of the adopted approximation. It is possible to find such a correspondence also for the GFs for more complicated operators, as spin or Hubbard operators where the diagram technique is much more complex.

In recent years, a consistent way was developed for truncating a system of equation of motion for the GFs based on projection operator method (POM) similar to the Mori [14] memory function (for references, see [15]). The POM was used by various authors (see, e.g., [16-20]) and in the most general way was formulated by Tserkovnikov [21,22]. In particular, it was shown in [17] that the POM for the two-time spin GF allows to reproduce results of the diagram technique for the spectrum of spin excitations in the ferromagnetic Heisenberg model. The method was also successfully used in studies of strongly anharmonic crystals close to the melting point [18]. An extensive study of the Hubbard model within the POM was performed by Mancini and Avella (for references see [19] and Chap. 3.7). Generally, it has been proved that the Mori-type projection technique for the time-dependent correlation functions provides reliable results in study spin dynamics, dense liquids, and phase transitions where strong correlations are essential as in SCES (for references see, e.g., [23]).

In the next section, we formulate a general POM in the theory of GFs. The method is applied to study superconductivity within the Hubbard model in Sect. 2 and to consider spin excitations in Sect. 3 in the t-J model.

6.2 Equation of Motion Method for Green Functions 6.2.1 General Formulation

We introduce the thermodynamic retarded (r), advanced (a), and causal (c) two-time Green functions (GFs) as defined by Zubarev [11]:

GAa(t - t') = {{A(t)|B(t')))r = -i0(t - t'){A(t)B(t') - ^B(t')A(t)), G5a(t - t') = {{A(t)|B(t')))a = i0(t' - t){A(t)B(t') - ^B(t')A(t)), GAa(t - t') = {{A(t)|B(t')))c = - { 7 A(t)B(t')), (6.2)

where the time-ordered product of operators is defined as 7°A(t) B(t') = 0(t -1') A(t)B(t') + ^0(t' - t)B(t')A(t). Here, the step-function 0(t) = 1, for t > 0 and 0(t) = 0 for t < 0, A(t) = exp(iHt)A exp(-iHt) is a time-dependent operator in the Heisenberg representation and {AB) = (1/Z)7rfexp(-PH)AB}, Z = 7rfexp(-PH)} is the statistical average (we set P = 1/7, kB = 1, „ = 1). The parameter ^ = +1 is taken for commutator GFs and ^ = -1 for anticommutator GFs: [A,B], = AB - ^BA .

In this notation, equations of motion for all types of GFs are the same and can be written as idt {{A(t)|B(t'))) = 1(t - t') {[A(0), B(0)]„) + {{[A(t), H]|B(t'))). (6.3)

The Fourier transformation for the GFs (6.2) and the respective time correlation functions read,

Gab (t - t') = — Gab (! e"i!(t"t0)d!, (6.4)

{B(t')A(t)) = — Jba (!) e"i!(t"t 0) d!, (6.5)

where Fourier transformation for the correlation function {A(t)B(t )) is similar to (6.5) with the spectral function JAB(!) = e_p!JBA(-!). The spectral representation for the retarded (advanced) GF in (6.2) can be written as

GrAaB(E) = ((A\B))E = — --jba! dm, e ! 0+ . (6.6)

The retarded GF GAB(E) is an analytic function in the upper half-plane of the a single analytic function can be introduced, GAB.E) = GAB(E) for ImE > 0

complex variable E, ImE > 0, while the advanced GF GaAB{E) is an analytic function in the lower half-plane of the complex variable E, ImE < 0 and therefore,


and Gab.E) = GaAB.E) for ImE <0 [10]. The analytical GF Gab.E) obeys the

dispersion relation

n J_oa z — m where P denotes the principal value of the integral. At the same time, the causal GF Gab(E) in (6.2) has a more complicated than (6.6) spectral representation, which shows that it cannot be analytically continued into the complex E plane at nonzero temperature, T > 0 (see [13]). In this respect, GCAB(E) is less convenient for application than the analytical function GAB.E). Moreover, the retarded commutator GFs have a simple physical meaning. They are directly related to the complex admittance, or the generalized susceptibility, xAB(r) = — hhA|B))! n=+l, which describes an influence of an external perturbation determined by the operator B on the average value of the dynamical variable A.

From the spectral representation (6.6) we obtain the relation between the spectral density of the correlation function and the GF:

Jba.r) = i {Gab.r C ie) — Gab.r — ie)} .ePr —

where the last relation holds for a real function JBA(rn) for operators A, B of the same parity with respect to time inversion. The spectral density (6.8) is related to the dynamical structure factor of a system which is measured in scattering experiments and can be used in calculation of the time-dependent correlation functions (6.5).

In study of collective excitations, such as spin or charge fluctuations, it is more convenient to consider the Kubo-Mori relaxation function (cf. [15,22]),

0AB.t — t') = ((A(t)IB(t'))) = —i0(t — t ')(A(t)IB(t')) 1

2n J-OO

Here, the Kubo-Mori scalar product is defined as,

The retarded commutator GF in (6.2) and the relaxation function (6.9) are coupled by the equation

Useful relations follow from (6.2) and (6.9)-(6.11):

((iA|B))ffi = ((A| - iB))! = ((A\B>>!, (6.12)

(iA|B) = (A|-iB) = ([A,B]>, (A|B) = —((A|B >>m=0, (6.13)

However, it should be pointed out that the relation (6.8) and (6.12), (6.13) can be used for the commutator GF for ergodic systems only. In those systems, the time-dependent correlation function (6.5) decays with time, limt!1(BA(t)> ! 0 (assuming that (B> = (A> = 0), and its spectral density is a regular function. For a nonergodic system, the spectral density can be written as,

where the first term is a regular function, while the second term is an irregular part. The latter can be calculated from the pole of the anticommutator GFs (6.2):

The nonergodic constant CAB determines the difference between the isothermal thermodynamic susceptibility xTAB = (A|B) and the Kubo (isolated) static susceptibility Xkb = —Gab(! = 0):

The nonergodic behavior occurs for a dynamical variable A coupled to integrals of motion, (AKn> ^ 0, where [Kn,H] = 0 [24].

6.2.2 Projection Technique for Green Functions

In this section, we formulate a general projection technique method in the equations of motion for the GFs (6.2) or the relaxation function (6.9). The method is based on the differentiation the GFs over two times, t and t', that enables to derive a Dyson-type equation with a self-energy similar to the memory function in the Mori projection technique [14].

We consider the GF (6.2) for arbitrary dynamical operators Ak,A£,

In a general case, the operators Ak and A+ are vectors and the GF is a matrix. Using (6.3), an equation of motion for the GF (6.17) reads, i dt <<Ak(t)|A+(t'))) = 1(t - t') <[Ak,Ak']„) C <<[Ak(t), H]|A+ (t'))). (6.18)

It is convenient to extract a linear term in the equation of motion for the original operator Ak in (6.18) by using the Mori-type projection technique:

The irreducible Zk )-operator is determined by the orthogonality condition,

([Zkir), A+]„ ) = (Zkir)A+ - rçA+Zf) = 0. (6.20)

This defines the frequency matrix,

Ekq = X<[ [Ak, H], Ak0]„) , Ikq = <[Ak,A?],). (6.21) k0

Using the Fourier transformation (6.4), the equation for the GF (6.17) can be written as,

Gk,k0(!) = Gg,(!) C £ Gg(!)/",<<Zjr)|A+))!, (6.22)

where we introduced the zero-order GF,

q which defines the excitation spectrum in the generalized mean-field approximation (GMFA).

Differentiating the many-particle GF {{Zqr)(t)|AC (t'))) over the second time t' and using the same projection procedure as above, the equation (6.22) can be written in the form,

Gk,k (!) = Gg, (!) + X Gg (!) Tq,q0 (!) Gq^, (!). (6.24)

We introduced here the scattering matrix,

Tkk (!) = £ /" <<Zf|(Zj))+))!/i7jt,. (6.25)

To obtain the Dyson equation, we define the self-energy matrix Xk,k'(!) by the equation,

Tfc,k' (!) = £k,k' (!) C X ^k,?.!)^, (!) T?0,k0 (!) , (6.26)

which shows that the self-energy matrix is a proper part of the scattering matrix which has no parts coupled by a single zero-order GF. An exclusion of the free (GMFA) GF in the definition of the self-energy (6.26) is equivalent to an introduction of the projected Liouvillian superoperator for the memory function in the original Mori technique [14]. Thus, we obtain an exact Dyson equation for the GF,

Gkk(!) = Gg,(!) C J2 Gk0)(!) (!) G?',k' (!), (6.27)

where the self-energy matrix, defined for the irreducible operators Zk ) = [Ak, H]? Ek,?A? , reads,

^k,k'(!) = X Ik? ((Zf)l(Z;i;))+»!proper)/?7>1k,• (6.28)

In comparison with a standard diagram technique, where the self-energy operator in the Dyson equation is written as a full vertex and a product of full GFs for excitations under study, the self-energy operator in the projection technique method is defined by the zeroth-order vertices and the full multiparticle GF which describes inelastic scattering of quasiparticles.

To consider higher order contributions to the self-energy, a general theory proposed by Tserkovnikov[21,22] can be used. In the theory, formally an exact system of equations is derived for the GFs (6.2) or (6.9) by differentiating them over time t and t'. For a sequence of operators Ai,A2,...A„ where Ai is the original dynamic operator for a physical quantity under study and the higher orders operators are given by time derivatives: /An = [An, H], an infinite system of coupled equations (in fact, identities) can be written:

!«A„|AC»!,n-i = (A»|A+) C {(/A»|A+) C ((/A»I - / A+»!,„}

x{A„|AC)-1((A„|AC))!;»—i; n = 1,2,..., (6.29) Here, the irreducible GFs of the nth order are coupled by the recurrence relation,

- {(A|B»+))!,»—i((A»|B+))—I»—i((A»|B+)>!,»—i, (6.30)

where ((A|AC))!;n = {{Am|B+))!;„ = 0, for m < n. The higher order operators An are also irreducible operators being orthogonal in respect to the scalar products (A»|A+):

(A | B + >n = (A | B + >n-l — (A|Bn+>n-l(An|Bn+>-^1(An|B + >n-1, (6.31)

where (A|B+>n = (Am|B+>n = 0, for m < n. In principle, within this approach, higher order corrections to the self-energy can be found. However, in many cases, the so-called mode-coupling approximation (MCA) for the lowest order self-energy (6.28) (equivalent to the noncrossing approximation in the diagram technique) can be used to obtain physically reasonable results. We discuss this approximation in the next sections for particular models.

6.3 Superconducting Pairing in the Hubbard Model

In this section, we apply the POM to study superconducting pairing in the Hubbard model (6.1). There are still hot debates whether the repulsion interaction in the Hubbard model could provide superconducting pairing and explain high-temperature superconductivity observed in cuprates. Our numerical calculations have definitely confirmed such a possibility [25-27] (see also [28-31]).

6.3.1 Hubbard Model

We consider the Hubbard model on a square lattice in a hole representation usually used in describing cuprate superconductors,

+ J2 ty {X/^Xf + X2aXf + a(X2NXy0a + H.c.)}, (6.32) i where X"^ = |ia>(i^| are the Hubbard operators (HOs) for the four states: an empty state |0>, a singly occupied state |a> with spin a/2 = (", #), a = ±1, a = —a, and a doubly occupied state |2> = | "#>. We denote the single-site repulsion energy by U and introduce E1 = e1 — ^ and E2 = 2E1 + U as the energy levels for the one-hole (with a reference energy e1) and the two-hole states, respectively. The model (6.32) can be used to study cuprate superconductors where, within the cell-cluster perturbation theory, the repulsion energy is defined by the charge-transfer gap in cuprates, U = (see, e.g., [32, 33]). In this case, the one-hole band is the d -like copper band, e1 = ed, and the two-hole band is the Zhang-Rice (ZR) p-d singlet band, e2 = ed + ep, [34]. We consider a strongly correlated limit of the Hubbard model, U ^ t, with an insulating state at half-filling (n = 1 ) when the atomic representation in terms of HOs is appropriate.

The bare electron dispersion is defined by the hopping parameter ty, which k-dependence is specified by the equation, t(k) = 4t y(k) + 4t' y'(k) + 4t'' y''(k),

where the hopping parameters are equal to t for the nearest neighbors and t', t'' for the second neighbors, which determine the bare (band) dispersion by the functions: y(k) = (1/2)(cos + cos ky), y'(k) = cos cos , and y''(k) = (1/2)(cos2kx + cos 2ky). In the cell-cluster perturbation theory, one can take t ' ' 0.4 eV [33]. The chemical potential ^ depends on the average hole occupation number, n = (N,), N, = J] X,™ + 2X,22. (6.34)

The spin operators in terms of HOs are defined as,

The HOs satisfy the completeness relation X,00 + X,"" + X## + X,22 = 1 and the multiplication rules Xf^X?"5 = I^X®5. From the latter follow, the commutation relations,

X0^ Xyi

The upper sign pertains to Fermi-type operators such as X,0ff which change the number of particles and the lower sign pertains to Bose-type operators, for example, the particle number operator N, in (6.34) or the spin operators Sf (6.35).

We emphasize that the Hubbard model (6.32) does not involve a dynamical coupling of electrons (holes) to fluctuations of spins or charges. Its role is played by the kinematic interaction caused by the non-Fermi nature of commutation relations (6.36), as already has been noted by Hubbard [8]. For example, the equation of motion for the HO X,ff2 has the form, i dXf2/ dt = [X(a2 , H ] = (Ei C U)X(a2 (6.37)

Bw = (X,22 C Xff) la0f + Xff la0f = (Ni/2 + SZ) 1f0f C Sf la0N , Bw = (Ni/2 C SZ) 1f0f - Sf la0f . (6.38)

Here, are Bose-like operators related to the particle number operator N, and the spin operators Sf (6.35).

6.3.2 Dyson Equation

To consider the superconducting pairing in the Hubbard model (6.32), we introduce the 4 x 4 matrix anticommutator GFs (6.2) [35]

where Xia and Xla = (X2a X°0 X" X^ ) are the four-component Nambu operators. Because of the two-subband nature of the model (6.32), the normal Gija and anomalous Fja components of the GF are 2 x 2 matrices which are coupled by the symmetry relations for the GFs [11].

To calculate the GF (6.39), we use the POM introduced above. As a result, we derive the following equation for the frequency matrix (6.21):

j = ({[Xia, H], X)a})Q~\ {A, B} = AB + BA, (6.40) „A A-( Q2 0

Here, t0 is the 2 x 2 unit matrix and in a paramagnetic state the coefficients Q2 = (X22 + Xaa) = n/2 and Qi = (X°0 + X~f ) = 1 — Q2 depend on the occupation number of holes (6.34) only. In the Q matrix, we neglected anomalous averages of the type (X02) which give no contribution to the d-wave pairing.

Frequency matrix (6.40) determines quasiparticle (QP) spectrum in the GMFA given by poles of the zeroth-order GF:

where i0 is 4 x 4 unit matrix. According to (6.27), the Dyson equation in the (q, r)-representation can be written as,

where the self-energy matrix (6.28) reads,

The system of equations (6.40)-(6.44) give an exact representation for the GF (6.39). To obtain a closed system of equations, we have to evaluate the multiparticle GF in the self-energy operator (6.44), which describes processes of inelastic scattering of electrons (holes) on charge and spin fluctuations due to kinematic interaction.

6.3.3 Mean-Field Approximation

The superconducting pairing in the Hubbard model already occurs in the GMFA and is caused by the kinetic exchange interaction as proposed by Anderson [36,37]. It is therefore reasonable to consider the GMFA described by the GF (6.42) separately. Using commutation relations for the HOs (6.36), we evaluate the frequency matrix (6.40),

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