Electronic spectrum and superconductivity in the t-J model on the honeycomb lattice

A microscopic theory of electronic spectrum and superconductivity within the t-J model on the honeycomb lattice is formulated. The Dyson equation for the normal and anomalous Green functions for the two-band model in terms of the Hubbard operators is derived by applying the Mori-type projection tec...

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Datum:	2018
1. Verfasser:	Plakida, N.M.
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Sprache:	English
Veröffentlicht:	Інститут фізики конденсованих систем НАН України 2018
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spelling	irk-123456789-1574592019-06-21T01:28:07Z Electronic spectrum and superconductivity in the t-J model on the honeycomb lattice Plakida, N.M. A microscopic theory of electronic spectrum and superconductivity within the t-J model on the honeycomb lattice is formulated. The Dyson equation for the normal and anomalous Green functions for the two-band model in terms of the Hubbard operators is derived by applying the Mori-type projection technique. The self-energy is evaluated in the self-consistent Born approximation for electron scattering on spin and charge fluctuations induced by the kinematical interaction for the Hubbard operators. Superconducting pairing mediated by the antiferromagnetic exchange and spin fluctuations is discussed. Сформульована мiкроскопiчна теорiя електронного спектра та надпровiдностi в рамках t-J моделi на стiльниковiй ґратцi. З допомогою технiки проекцiйних операторiв Морi отримано рiвняння Дайсона для нормальних i аномальних функцiй Ґрiна для двозонної моделi в термiнах операторiв Хаббарда. Власноенергетична частина розрахована у самоузгодженому борнiвському наближеннi для розсiяння електронiв на спiнових i зарядових флуктуацiях iндукованих кiнематичною взаємодiєю для операторiв Хаббарда. Обговорюється можливiсть надпровiдного спарювання через антиферомагнiтний обмiн та спiновi флуктуацiї. Ключовi 2018 Article Electronic spectrum and superconductivity in the t-J model on the honeycomb lattice / N.M. Plakida // Condensed Matter Physics. — 2018. — Т. 21, № 3. — С. 33706: 1–11. — Бібліогр.: 25 назв. — англ. 1607-324X PACS: 74.20.Mn, 71.27.+a, 71.10.Fd, 74.72.-h DOI:10.5488/CMP.21.33706 arXiv:1809.10287 http://dspace.nbuv.gov.ua/handle/123456789/157459 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	A microscopic theory of electronic spectrum and superconductivity within the t-J model on the honeycomb lattice is formulated. The Dyson equation for the normal and anomalous Green functions for the two-band model in terms of the Hubbard operators is derived by applying the Mori-type projection technique. The self-energy is evaluated in the self-consistent Born approximation for electron scattering on spin and charge fluctuations induced by the kinematical interaction for the Hubbard operators. Superconducting pairing mediated by the antiferromagnetic exchange and spin fluctuations is discussed.
format	Article
author	Plakida, N.M.
spellingShingle	Plakida, N.M. Electronic spectrum and superconductivity in the t-J model on the honeycomb lattice Condensed Matter Physics
author_facet	Plakida, N.M.
author_sort	Plakida, N.M.
title	Electronic spectrum and superconductivity in the t-J model on the honeycomb lattice
title_short	Electronic spectrum and superconductivity in the t-J model on the honeycomb lattice
title_full	Electronic spectrum and superconductivity in the t-J model on the honeycomb lattice
title_fullStr	Electronic spectrum and superconductivity in the t-J model on the honeycomb lattice
title_full_unstemmed	Electronic spectrum and superconductivity in the t-J model on the honeycomb lattice
title_sort	electronic spectrum and superconductivity in the t-j model on the honeycomb lattice
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2018
url	http://dspace.nbuv.gov.ua/handle/123456789/157459
citation_txt	Electronic spectrum and superconductivity in the t-J model on the honeycomb lattice / N.M. Plakida // Condensed Matter Physics. — 2018. — Т. 21, № 3. — С. 33706: 1–11. — Бібліогр.: 25 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT plakidanm electronicspectrumandsuperconductivityinthetjmodelonthehoneycomblattice
first_indexed	2025-07-14T09:53:11Z
last_indexed	2025-07-14T09:53:11Z
_version_	1837615590915702784
fulltext	Condensed Matter Physics, 2018, Vol. 21, No 3, 33706: 1–11 DOI: 10.5488/CMP.21.33706 http://www.icmp.lviv.ua/journal Electronic spectrum and superconductivity in the t-J model on the honeycomb lattice∗ N.M. Plakida Joint Institute for Nuclear Research, 141980 Dubna, Russia Received June 1, 2018, in final form July 27, 2018 A microscopic theory of electronic spectrum and superconductivity within the t-J model on the honeycomb lat- tice is formulated. The Dyson equation for the normal and anomalous Green functions for the two-band model in terms of the Hubbard operators is derived by applying the Mori-type projection technique. The self-energy is evaluated in the self-consistent Born approximation for electron scattering on spin and charge fluctuations induced by the kinematical interaction for the Hubbard operators. Superconducting pairing mediated by the antiferromagnetic exchange and spin fluctuations is discussed. Key words: strong electron correlations, t-J model, honeycomb lattice, superconductivity PACS: 74.20.Mn, 71.27.+a, 71.10.Fd, 74.72.-h 1. Introduction One of the crucial issues in the current theory of condensed matter is the study of superconductivity in strongly correlated electronic systems, as in the cuprate high-temperature superconductors (see, e.g., [1]). The basic model commonly used in these studies is the Hubbard model which is specified by two parameters: the single-electron hopping matrix element t and the single-site Coulomb energy U. A rigorous analytical treatment of the model is based on the Hubbard operator (HO) technique [2] since in this representation the local constraint of no double occupancy of any lattice site is rigorously implemented by the Hubbard operator algebra. A superconduction pairing due to the kinematical interaction in the Hubbard model in the limit of strong electron correlations (U → ∞) was first considered by Zaitsev and Ivanov [3] who used a diagram technique for the HOs. Subsequently, superconducting pairing in the Hubbard model was studied by Plakida and Stasyuk [6] by applying the equation of motion method for the thermodynamic Green functions (GFs) [7, 8]. In the later studies [9, 10] of the t-J model within the Mori-type projection technique [11, 12], a formula for the superconduction temperature similar to [3] was obtained in the mean-field approximation (MFA). A self-consistent solution of Dyson equations beyond theMFA for normal and anomalous GFs and respective self-energies within the t-J model was performed in [13] that confirmed the results of the previous studies in MFA [9, 10]. Detailed studies of the extended Hubbardmodel on the square lattice were performed in [14, 15]. Taking into account an intersite Coulomb repulsion and electron-phonon interaction, a microscopic theory of superconductivity in cuprates was formulated. It was proved that the spin-fluctuation pairing mechanism induced by the strong kinematical interaction is the leading interaction which results in high-temperature superconductivity. In recent years, the two-dimensional carbon honeycomb lattice, the graphene, has been extensively studied due its peculiar electronic properties (for a review see [16, 17]). Studies of the graphene beyond the simple model of noninteracting electrons by taking into account the Coulomb interaction reveal a rich phase diagram with phase transitions to the antiferromagnetic (AF) state, spin-density wave (SDW), charge-density wave (CDW), and nonconventional superconductivity (SC). Superconducting phase transitions in the Hubbard model on the honeycomb lattice have been considered in several ∗The paper is devoted to the 80-th birthday of Professor I.V. Stasyuk. This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. 33706-1 https://doi.org/10.5488/CMP.21.33706 http://www.icmp.lviv.ua/journal http://creativecommons.org/licenses/by/4.0/ N.M. Plakida publications. The renormalization group approach was used in [18] to study phase transitions in the extended Hubbard model with a moderate on-site interaction U. Close to half-filling, the SDW or CDW orders occur, while for a large doping, f -wave triplet-pairing and d + id-wave singlet-pairing appear. Using the dynamic cluster approximation for the Hubbard model with U/t = 2 − 6, a transition from the d+ id-wave singlet pairing at weak coupling to the p-wave triplet pairing at larger coupling was observed in [19]. In the limit of strong correlations,U � t, two conduction bands of electrons in the honeycomb lattice split into the singly- and doubly-occupied Hubbard subbands. In this limit, the Hubbard model can be reduced to the two-band t-J model for the projected electron operators. A detailed study of the t-J model on the honeycomb lattice was presented in [20]. The ground-state energy and the staggered magnetization in the AF phase as function of doping δ have been calculated using the Grassmann tensor product state approach, exact diagonalization and density-matrix renormalization methods. The occurrence of the time-reversal symmetry breaking d + id-wave SC at large doping was found. Moreover, a coexisting of the SC and AF order was observed for low doping, 0 < δ < 0.1, where the triplet pairing is induced. In the above cited papers, mostly the phase diagram of the models with the Coulomb interaction on the honeycomb lattice was studied at zero temperature by numerical methods. To investigate temperature and doping dependence of the phase diagram and, in particular, superconduction phase transition analytical methods should be used. The honeycomb Heisenberg model at half-filling over the whole temperature region both in the AF and paramagnetic phases was studied in [21]. The electronic and spin-fluctuation spectra in this model were studied in [22] using the generalized mean-field approximation (GMFA). In the present paper, we formulate a microscopic theory of superconductivity within the two-band t-J model on the honeycomb lattice. We derive Dyson equation for the normal and anomalous matrix GFs with the self-energy evaluated in the self-consistent Born approximation (SCBA) as for the one-band t-J model in [13] and the Hubbard model in [14, 15] on the square lattice. Electronic spectrum and superconductivity is considered in the MFA. In section 2, we formulate the t-J model in terms of Hubbard operators. The Dyson equation is derived in section 3. The results of calculation of the electronic spectrum and the superconduction gap function are presented in section 4. The conclusion can be found in section 5. 2. The t-J model We consider the t-J on the honeycomb lattice. The honeycomb lattice is bipartite with two triangular sublattices A and B. Each of the N sites on the A sublattice is connected to three nearest-neighbor (nn) sites belonging to the B sublattice by vectors δα, and N sites on B are connected to A by vectors −δα: δ1 = a0 2 (√ 3,−1 ) , δ2 = − a0 2 (√ 3, 1 ) , δ3 = a0(0, 1). (2.1) The basis vectors are a1 = δ3−δ2 = (a0/2)( √ 3, 3) and a2 = δ3−δ1 = (a0/2)(− √ 3, 3), the lattice constant is a = \|a1 \| = \|a2 \| = √ 3a0, where a0 is the nn distance; hereafter we put a0 = 1. The reciprocal lattice vectors are k1 = (2π/3)( √ 3, 1) and k2 = (2π/3)(− √ 3, 1). In conventional notation, the t-J model reads: H = −t ∑ 〈i, j 〉σ ã+i,σ ãj,σ − µ ∑ i,σ ni,σ + J 2 ∑ 〈i, j 〉 ( Si Sj − 1 4 ni nj ) , (2.2) where for the two-sublattice representation A, B of the site indices on the honeycomb lattice, a shorter notation iα→ i is used. The projected electron operators in the singly occupied Hubbard subband in the model (2.2) are defined as ã+i,σ = a+i,σ(1 − ni,σ̄) and ãiσ = aiσ(1 − ni,σ̄) for creation and annihilation of an electron with spin σ/2 (σ = ±1, σ̄ = −σ) , ni,σ = ã+i,σ ãi,σ , and ni = ∑ σ ni,σ . Here, t is the nn electron hopping energy and J = 4t2/U is the nn AF exchange interaction for electronic spins Si . To take into account, on a rigorous basis, the no-double-occupancy constraint for the projected electron operators ã+i,σ , we employ the Hubbard operator (HO) technique [2]. The HOs Xnm i = \|i, n〉〈i,m\| 33706-2 t-J model on the honeycomb lattice determine transitions between three possible states on a lattice site i: \|i, n〉 = \|i, 0〉 and \|i, σ〉 for an empty site and for a singly occupied site by an electron with spin σ/2, respectively. The electron number operator and the spin operators in terms of HOs are defined as ni = ∑ σ Xσσ i = X++i + X−−i , (2.3) Sσi = Xσσ̄ i , Sz i = (σ/2) [X σσ i − X σ̄σ̄ i ]. (2.4) The completeness relation for the HOs, X00 i + ∑ σ Xσσ i = 1, rigorously preserves the constraint of no- double-occupancy for a quantum state \|i, n〉 on any lattice site i. From the multiplication rule Xnm i Xkl i = δmkXnl i for HOs, there follow the commutation relations:[ Xnm i , Xkl j ] ± = δi j ( δmkXnl i ± δnlX km i ) . (2.5) The upper sign refers to Fermi-type operators such as X0σ i , while the lower sign refers to Bose-type operators such as ni (2.3) or the spin operators (2.4). Using the Hubbard operator representation for ã+iσ = Xσ0 i , ãjσ = X0σ j and equations (2.3) and (2.4), we write the Hamiltonian of the t-J model (2.2) in the form: H = −t ∑ 〈i, j 〉σ Xσ0 i X0σ j − µ ∑ iσ Xσσ i + J 4 ∑ 〈i, j 〉σ ( Xσσ̄ i X σ̄σ j − Xσσ i X σ̄σ̄ j ) . (2.6) The chemical potential µ depends on the average electron occupation number n = nA = nB , nα = 1 N ∑ i,σ 〈 niα,σ〉 , (2.7) where N is the number of unit cells and 〈...〉 denotes the statistical average with the Hamiltonian (2.6). 3. Dyson equation To consider superconductivity within the model (2.6), we introduce the anticommutator two-time matrix Green function (GF) [7, 8] Gi jσ(t − t ′) = −iθ(t − t ′)〈{X̂iσ(t), X̂†jσ(t ′)}〉 ≡ 〈〈X̂iσ(t), X̂†jσ(t ′)〉〉 , (3.1) where X(t) = eiHt Xe−iHt and θ(x) is the Heaviside function. Here, we use Nambu notation and introduce the Hubbard operators in the two-sublattice representation X̂iσ and X̂†iσ where X̂iσ = ©« X0σ iA X0σ iB X σ̄0 iA X σ̄0 iB ª®®®®®¬ , X̂†jσ = ( Xσ0 jA Xσ0 jB X0σ̄ jA X0σ̄ jB ) . (3.2) The Fourier representation in (k, ω)-space is defined by Gi jσ(t − t ′) = ∞∫ −∞ dω 2π e−iω(t−t ′) 1 N ∑ k eik(ri−r j )Gσ(k, ω). (3.3) The 4 × 4 matrix GF (3.1) can be written as Gi jσ(ω) = 〈〈X̂iσ \| X̂†jσ〉〉ω = ( Ĝi jσ(ω) F̂i jσ(ω) F̂†i jσ(ω) − Ĝ jiσ̄(−ω) ) , (3.4) 33706-3 N.M. Plakida where the normal Ĝi jσ and anomalous F̂i jσ components of the GF are 2× 2 matrices, which are coupled by the symmetry relations for the anticommutator retarded GF [7, 8]. To calculate the GF (3.1), we use the equation of motion method. Differentiating the GF with respect to time t, its Fourier representation leads to the equation ωGi jσ(ω) = δi jQ + 〈〈[X̂iσ,H] , X̂†jσ〉〉ω , (3.5) where Q = 〈{X̂iσ, X̂†iσ}〉 = τ̃0Q . Here, τ̃0 is the 4 × 4 unit matrix and in a paramagnetic state, the coefficient Q = 〈X00 iα + Xσσ iα 〉 = 1 − nα/2 depends on the occupation number of electrons (2.7) only. For a system of strongly correlated electrons as in the t-J model, there is no well-defined quasi- particle (QP) excitations specified by zeroth-order kinetic energy. Therefore, it is convenient to choose the mean-field contribution to the energy of QPs in the equations of motion (3.5) as the zeroth-order QP energy. To identify this contribution, we use the projection operator method developed for the GF [12]. To this end, we write the operator Ẑiσ = [X̂iσ,H] in (3.5) as a sum of the linear part, proportional to the original operator X̂iσ , and the irreducible part Ẑ (ir)iσ orthogonal to X̂iσ: Ẑiσ = [X̂iσ,H] = ∑ l Eilσ X̂lσ + Ẑ (ir)iσ . (3.6) The orthogonality condition 〈{Ẑ (ir)iσ , X̂†jσ}〉 = 〈Ẑ (ir) iσ X̂†jσ + X̂†jσ Ẑ (ir)iσ 〉 = 0 determines the linear part, the frequency matrix: Ei jσ = 〈{[X̂iσ,H], X̂†jσ}〉Q−1 = ( Êi j ∆̂i jσ ∆̂∗jiσ − Êji ) . (3.7) Frequency matrix (3.7) determines QP spectrum Êi j and the gap function ∆̂i jσ in the GMFA. The corresponding zeroth-order GF in the Fourier representation reads G0 σ(k, ω) = [ ωτ̃0 − Eσ(k)]−1Q , E(k) = ( Ê(k) ∆̂σ(k) ∆̂∗σ(k) − Ê(k) ) . (3.8) Differentiating the multiparticle GF 〈〈Ẑiσ(t) , X̂†jσ(t ′)〉〉 in (3.5) with respect to the second time t ′ and using the same projection procedure as in (3.6) leads to the Dyson equation for the GF (3.1). In the (k, ω)-representation, the Dyson equation reads [Gσ(k, ω)]−1 = [G0 σ(k, ω) ]−1 − Σ̃σ(k, ω). (3.9) The self-energy operator Σ̃σ(k, ω) is defined by the proper part of the scattering matrix which has no parts connected by the zeroth order GF (3.8): Σ̃σ(k, ω) = Q−1〈〈Ẑ (ir)kσ \| Ẑ (ir)† kσ 〉〉 (prop) ω Q−1. (3.10) The self-energy operator (3.10) can be written in the same form as GF (3.4): Σ̃i jσ(ω) = Q−1 ( M̂i j(ω) Φ̂i jσ(ω) Φ̂ † i jσ(ω) − M̂ji(−ω) ) Q−1 . (3.11) The normal M̂ and anomalous (pair) Φ̂ components of the self-energy operator (3.11) are given by the 2 × 2 matrices [see (3.22), (3.23)]. The system of equations for the (4 × 4) matrix GF (3.4) and the self-energy (3.11) can be reduced to a system of equations for the normal Ĝσ(k, ω) and anomalous F̂σ(k, ω) (2 × 2) matrix components. By using representations for the frequency matrix (3.7) and the self-energy (3.11), we derive the following system of matrix equations for them: Ĝσ(k, ω) = [ ĜN (k, ω)−1 + ϕ̂σ(k, ω) ĜN (k,−ω) ϕ̂∗σ(k, ω) ]−1 Q , (3.12) F̂†σ(k, ω) = −ĜN (k,−ω) ϕ̂†σ(k, ω) Ĝσ(k, ω). (3.13) 33706-4 t-J model on the honeycomb lattice In (3.12), we introduced the normal state matrix GF and the matrix superconduction gap function: ĜN (k, ω) = [ ωτ̂0 − ε̂(k) − M̂(k, ω)/Q̂ ]−1 , (3.14) ϕ̂σ(k, ω) = ∆̂σ(k) + Φ̂σ(k, ω)/Q̂. (3.15) To obtain a closed system of equations, we should evaluate the multiparticle GF in the self-energy operator (3.11) which describes the processes of inelastic scattering of electrons on charge and spin fluctuations due to the kinematic interaction. 3.1. Mean-field approximation The superconducting pairing in the Hubbard model already occurs in the MFA and is caused by the kinetic exchange interaction as proposed by Anderson [24, 25]. It is, therefore, reasonable to consider the MFA described by zeroth-order GF (3.8) separately. Using commutation relations for Hubbard operators (2.5), we evaluate the frequency matrix (3.7). The matrix Ê(k) determines the QP spectrum in the normal phase (see [22]): Ê(k) = 1 N ∑ i, j exp[ik(ri − rj)]Êi j = ( εA(k) εAB(k) εBA(k) εB(k) ) , εA(k) = 〈{[X0σ kA ,H], Xσ0 kA }〉 Q−1, εB(k) = 〈{[X0σ kB ,H], Xσ0 kB }〉 Q−1, εAB(k) = 〈{[X0σ kA ,H], Xσ0 kB }〉 Q−1, εBA(k) = (εAB(k))†. (3.16) The solution of the matrix equation for the zero-order GF (3.8) in the normal state reads: Ĝ0 N (k, ω) = ( G0 AA(k, ω) G0 AB(k, ω) G0 BA(k, ω) G0 BB(k, ω) ) = Q D(k, ω) ( ω − εB(k) εAB(k) εAB(k)∗ ω − εA(k) ) , (3.17) where D(k, ω) = [ω − εA(k)][ω − εB(k)] − \|εAB(k)\|2 = [ω − ε+(k)][ω − ε−(k)]. (3.18) The electronic spectrum has two branches: ε±(k) = 1 2 [εA(k) + εB(k)] ± 1 2 { [εA(k) − εB(k)]2 + 4\|εAB(k)\|2 }1/2 = ε(k) ± \|εAB(k)\| , (3.19) where we take into account the fact that in the paramagnetic state, the sublattices are equivalent and, therefore, εA(k) = εB(k) = ε(k). We calculate the spectrum and consider the Fermi surface in the normal state in the next section. Now, we evaluate the anomalous component ∆̂i jσ of the matrix (3.7), which determines the super- conducting gap. Similar to the normal-state frequency matrix (3.16), we obtain the representation ∆̂σ(k) = 1 N ∑ i, j exp[ik(ri − rj)] ( ∆i j,Aσ ∆i j,ABσ ∆i j,BAσ ∆i j,Bσ ) = ( ∆Aσ(k) ∆ABσ(k) ∆BAσ(k) ∆Bσ(k) ) , ∆Aσ(k) = 〈{[X0σ kA ,H], X0σ̄ kA }〉 Q−1, ∆Bσ(k) = 〈{[X0σ kB ,H], X0σ̄ kB }〉 Q−1, ∆ABσ(k) = 〈{[X0σ kA ,H], X0σ̄ kB }〉 Q−1, ∆BAσ(k) = ∆†ABσ(k). (3.20) Diagonalization of the gap matrix (3.20) shows a two-gap structure: ∆ ± σ(k) = 1 2 [ ∆Aσ(k) + ∆Bσ(k) ] ± 1 2 { [∆Aσ(k) − ∆Bσ(k)]2 + 4\|εAB(k)\|2 }1/2 = ∆σ(k) ± \|∆ABσ(k)\| , (3.21) where we take into account the fact that in the paramagnetic state, the sublattices are equivalent and, therefore, ∆Aσ(k) = ∆Bσ(k) = ∆σ(k). We consider the gap components (3.20) in the next section. 33706-5 N.M. Plakida 3.2. Self-energy operator The normal M̂ and anomalous (pair) Φ̂ components of the self-energy operator (3.11) are given by the 2 × 2 matrices: M̂i j(ω) = 〈〈( [X̃0σ iA ,H] [X̃0σ iB ,H] ) \| ([H, X̃σ0 jA ] [H, X̃σ0 jB ]) 〉〉(pp) ω , (3.22) Φ̂i jσ(ω) = 〈〈( [X̃0σ iA ,H] [X̃0σ iB ,H] ) \| ([H, X̃0σ̄ jA ][H, X̃0σ̄ jB ]) 〉〉(pp) ω , (3.23) where [�Xn,m iα ,H] are the irreducible parts of the commutators determined by equation (3.6). Taking into account equations for the [X0σ iα ,H] operators, we obtain the following representation for the matrix elements of the normal self-energy Mi jA(ω) = t2 ∑ l,σ′ ∑ k,σ′′ 〈〈B̃σσ ′ iA X0σ′ lB \| Xσ′′0 kB B̃σ ′′σ jA 〉〉 (pp) ω + J2 4 ∑ l,σ′ ∑ k,σ′′ 〈〈B̃σσ ′ lB X0σ′ iA \| Xσ′′0 jA B̃σ ′′σ kB 〉〉 (pp) ω , (3.24) Mi jAB(ω) = t2 ∑ l,σ′ ∑ k,σ′′ 〈〈B̃σσ ′ iA X0σ′ lB \| Xσ′′0 kA B̃σ ′′σ jB 〉〉 (pp) ω + J2 4 ∑ l,σ′ ∑ k,σ′′ 〈〈B̃σσ ′ lB X0σ′ iA \| Xσ′′0 jB B̃σ ′′σ kA 〉〉 (pp) ω , (3.25) where B̃σσ ′ iα = (Sz iα − niα/2) δσ,σ′ + Xσ′σ iα δσ′,σ̄ , (B̃σσ ′ iα ) † = B̃σ ′σ iα . (3.26) In (3.24), (3.25), we take into account only the diagonal in respect to the interaction terms. Using equations for the operators [X σ̄0 iα ,H], a similar representation is derived for the anomalous matrix elements (3.23): Φi jA(ω) = 〈〈[X̃0σ iA ,H] \| [H, X̃0σ̄ jA ]〉〉 (pp) ω , (3.27) Φi jAB(ω) = 〈〈[X̃0σ iA ,H] \| [H, X̃0σ̄ jB ]〉〉 (pp) ω . (3.28) To calculate the self-energy matrix elements in (3.24), (3.25), and (3.27), (3.28), we use the SCBA for the multiparticle GFs. For this, it is convenient to write the self-energy in terms of time-dependent correlation functions. In particular, for the normal component (3.24), we have the spectral representation: Mi jA(ω) = ∑ l,σ′ ∑ k,σ′′ 1 2π ∞∫ −∞ dz e βz + 1 (ω − z) ∞∫ −∞ dteizt × [ t2〈Xσ′′0 kB B̃σ ′′σ jA B̃σσ ′ iA (t)X 0σ′ lB (t)〉 + J2 4 〈Xσ′′0 jA B̃σ ′′σ kB B̃σσ ′ lB (t)X 0σ′ iA (t)〉 ] . (3.29) In the SCBA, a propagation of excitations described by the Fermi-like operators X̂σ0 i and the Bose-like operators B̃σ ′σ j for i , j is assumed to be independent. Therefore, the corresponding time-dependent multiparticle correlation functions can be written as a product of fermionic and bosonic correlation functions, 〈Xσ′′0 kB B̃σ ′′σ jA B̃σσ ′ iA (t)X 0σ′ lB (t)〉 = δσσ′ 〈X σ′0 kB X0σ′ lB (t)〉〈B̃ σ′σ jA B̃σσ ′ iA (t)〉. (3.30) 33706-6 t-J model on the honeycomb lattice The time-dependent correlation functions in (3.30) are calculated self-consistently using the correspond- ing GFs, as e.g., 〈Xσ′0 kB X0σ′ jB (t)〉 = ∞∫ −∞ dωn(ω)e−iωt [−(1/π)] Im〈〈X0σ′ jB \|X σ′0 kB 〉〉ω , (3.31) 〈B̃σ ′σ jA B̃σσ ′ iA (t)〉 = ∞∫ −∞ dωN(ω)e−iωt [−(1/π)] Im〈〈B̃σσ ′ iA \|B̃ σ′σ jA 〉〉ω , (3.32) where n(ω) and N(ω) are the Fermi- and Bose-function, respectively. Integration over time t in (3.29) yields Mi jA(ω) = ∑ l,k ∑ σ′ ∫ ∞∫ −∞ dω1dω2 π2 1 − n(ω1) + N(ω2) ω − ω1 − ω2 × ( t2Im〈〈X0σ′ lB \|X σ′0 kB 〉〉ω1 Im〈〈B̃σσ ′ iA \|B̃ σ′σ jA 〉〉ω2 + J2 4 Im〈〈X0σ′ iA \|X σ′0 jB 〉〉ω1 Im〈〈B̃σσ ′ lB \|B̃ σ′σ kA 〉〉ω2 ) . (3.33) Taking into account the definition of the bosonic operator (3.26), the bosonic GFs in these equations can be written as 〈〈B̃σσ ′ iA \|B̃ σ′σ jA 〉〉ω = 1 4 〈〈niA \|njA〉〉ω δσ′σ + 〈〈Sz iA \|S z jA〉〉ω δσ′σ + 〈〈X σ̄σ iA \|X σσ̄ jA 〉〉ω δσ′σ̄ . (3.34) After summation over σ′ in equation (3.33) for the bosonic GF (3.34) and taking into account that the normal GF in the paramagnetic state is independent of spin, the spin-fluctuation contribution can be written in the form: 〈〈Sz iA \|S z jA〉〉ω + 〈〈X σ̄σ iA \|X σσ̄ jA 〉〉ω = 〈〈SiA \|SjA〉〉ω ≡ −χi j,A(ω). (3.35) Introducing the k-representation for the GFs in the self-energy (3.33), we obtain the expression: MA(k, ω) = 1 N ∑ q +∞∫ −∞ dz 1 π +∞∫ −∞ dΩ 1 + N(Ω) − n(z) ω − z −Ω × { t2 [−(1/π)] Im〈〈X0σ Aq \|X σ0 Aq 〉〉z ImχA(k − q,Ω) + J2 4 [−(1/π)] Im〈〈X0σ Aq \|X σ0 Bq 〉〉zImχBA(k − q,Ω) } . (3.36) The SCBA for Mi jAB(ω) (3.25) and for the anomalous self-energies (3.27), (3.28) gives similar results. 4. Results To find out the normal state GF (3.14) beyond theMFA, the self-energy matrix elements (3.24), (3.25) should be calculated. Similar calculations of the anomalous self-energy (3.27), (3.28) give an equation for the frequency dependent gap (3.15). However, implementation of this program demands complicated self-consistent numerical computations for the two-band t-J model which are much more complicated than similar calculations performed for the one-band t-J model on the square lattice in [13]. Therefore, in the present paper, we restrict our consideration only to the results in MFA and study the complete system of equations later. 33706-7 N.M. Plakida 4.1. Electronic spectrum Calculation of the matrix elements (3.16) gives the following results: ε(k) = −µ + 3t Q D1 − 3J 4 nα + 3J 2Q C1 ≡ −µ̃ , (4.1) where we introduce the nn correlation functions for electrons and spins, D1 = 〈Xσ0 iA X0σ i+δ1,B〉 , C1 = 〈Sz iASz i+δ1,B〉. (4.2) For the off-diagonal energy, we have: εAB(k) = −̃t γ1(k) , t̃ = tQ ( 1 + 3C1 2Q2 ) + J D1 2Q , (4.3) where γ1(k) = ∑ α exp(ikδα) and \|γ1(k)\|2 = 1 + 4 cos( √ 3kx/2)[cos( √ 3kx/2) + cos(3ky/2)]. Thus, the electronic spectrum has two branches ε±(k) = −µ ± t̃ \|γ1(k)\| , (4.4) which is similar to the spectrum of the graphene (see, e.g., reference [16]) but with the renormalized chemical potential µ̃ due to strong correlations, equation (4.1), and the hopping parameter t̃ , equa- tion (4.3). Therefore, the spectrum shows a cone-type behaviour at Dirac points K and K ′ at the corners of the graphene Brillouin zone (BZ) as shown in figure 1. In figure 2, the BZ and the Fermi surfaces (FS) are shown for holes at the electronic occupation numbers n = 0.95, 0.76, and 0.7. At n . 1, the hole FS is small and centered at the Γ point. With a decreasing n, the FS becomes larger, and at some characteristic value n0 = 0.76, the FS touches the BZ at M-points. At a larger hole doping, six pockets centered at the K-points emerge which shrink to points for the half-filled band at n = 2/3. For the diagonal GF, we have Gαα,σ(k, ω) = Q 2 [ 1 ε+(k) − ω + 1 ε−(k) − ω ] . (4.5) The average occupation number of electrons in one sublattice is equal to nασ(k) = 〈Xσσ kα 〉 = 〈X σ0 kα X0σ kα 〉 = ∞∫ −∞ dω eω/T + 1 [ − 1 π ] ImGαα,σ(k, ω) , (4.6) �� 0� � �� Figure 1. Electronic spectrum ε±(k) in units of t̃ at µ̃ = 0. 33706-8 t-J model on the honeycomb lattice � �6 � �� Figure 2. Brillouin zone (bold) and hole Fermi surface for n = 0.95 (thin solid), 0.76 (dashed), and 0.7 (dotted). which yields in MFA nα,σ(k) = Q 1 2 [N(ε+(k)) + N(ε−(k))], (4.7) where N(ε±(k)) = {exp[ε±(k)/T] + 1}−1, and the average number of electrons in one sublattice is given by nα = 2 N ∑ k nασ(k) (4.8) with nα 6 1. For the off-diagonal GF in equation (3.10), we obtain GAB,σ(k, ω) = Q εAB(k) 2 \|εAB(k)\| [ 1 ε+(k) − ω − 1 ε−(k) − ω ] . (4.9) For the nn correlation function, we get 〈Xσ0 kB X0σ kA 〉 = Q γ1(k) 2\|γ1(k)\| [N(ε−(k)) − N(ε+(k))]. (4.10) These results can be used in calculating the self-energy in the normal state (3.36). 4.2. Gap equation The matrix elements of the gap function (3.20) are given by ∆i j,Aσ = 〈{[X0σ iA ,H], X0σ̄ jA }〉 Q−1, ∆i j,ABσ = 〈{[X0σ iA ,H], X0σ̄ jB }〉 Q−1. (4.11) Using the commutation relations for the HOs, for these functions we obtain the following representation: ∆i j,Aσ = −t ∑ k,σ′ 〈{Bσσ ′ iA X0σ′ kB , X0σ̄ jA }〉Q −1 + J 2 ∑ k,σ′ 〈{(Bσσ ′ kB − δσ,σ′)X 0σ′ iA , X0σ̄ jA }〉Q −1 = −tδi, j ∑ k 〈X0σ iA X0σ̄ kB − X0σ̄ iA X0σ kB 〉Q −1 = δi, j2t ∑ δα 〈X0σ̄ iA X0σ i+δαB〉Q −1, (4.12) ∆i j,ABσ = J 2 ∑ k,σ′ δk, j ( δσ′,σ̄ 〈X0σ jB X0σ̄ iA 〉 − δσ′,σ 〈X 0σ̄ jB X0σ iA 〉 ) Q−1 = −J 〈X0σ̄ iA X0σ jB 〉 Q−1, (4.13) where the symmetry relations were used: 〈X0σ jB X0σ̄ iA 〉 = −〈X 0σ̄ iA X0σ jB 〉 and 〈X 0σ̄ jB X0σ iA 〉 = −〈X 0σ jB X0σ̄ iA 〉 . 33706-9 N.M. Plakida Fourier transformation results in the gap function components: ∆Aσ(k) = 2t ∑ δα 〈X0σ̄ iA X0σ i+δαB〉Q −1, (4.14) ∆ABσ(k) = −J ∑ δα exp[ikδα] 〈X0σ̄ iA X0σ i+δαB〉Q −1, (4.15) where δα = δ1, δ2, δ3. Introducing the bond-dependent anomalous correlation functions FαABσ = 〈X 0σ̄ iA X0σ i+δαB〉 , (4.16) the gaps can be written as ∆Aσ = 2t ∑ δα FαABσQ−1, ∆ABσ(k) = −J ∑ δα exp[ikδα] FαABσQ−1. (4.17) Here, the nn correlation function (4.16) is calculated using the spectral representation: 〈X0σ̄ iA X0σ jB 〉 = ∞∫ −∞ dωn(ω)[−(1/π)] Im〈〈X0σ jB \|X 0σ̄ iA 〉〉ω . (4.18) For the bond-independent correlation functions (s-wave pairing), FαABσ = FABσ , we have the results similar to the normal state energy: ∆Aσ = 6tFABσ Q−1, ∆ABσ(k) = −Jγ(k)FABσ Q−1. (4.19) Using the GMFA for the anomalous GF in (4.18), we can derive the Bardeen-Cooper-Schrieffer (BCS) type equations as was proposed in several publications for electrons with the Dirac spectrum. In particular, in [23], the s-wave BCS model was considered with a phenomenological coupling constant. In the case of the two-subband honeycomb lattice, we expect to obtain a more complicated pairing symmetry, e.g., d + id singlet pairing or p triplet pairing as proposed in [20]. In the case of the t-J on the square lattice in [13], we have also found a more complicated d-wave pairing induced by spin-fluctuations. A self-consistent solution of the gap equations (4.17) with microscopic coupling constants and anomalous GF (3.13) and a resulting equation for determining the superconduction transition temperature will be considered in a subsequent publication. 5. Conclusions In the present paper, a microscopic theory of the electron spectrum and superconductivity within the two-band t-J model for strongly correlated electrons on the honeycomb lattice is presented. Using the projection operator method for thermodynamic GFs, we derived a self-consistent system of equations for the matrix GF and the self-energy in the SCBA. The latter is similar to the Migdal-Eliasberg strong- coupling theory for the electron-phonon system. In the present paper, we consider only the results obtained in the MFA. However, as it was pointed out in the microscopic theory of superconductivity formulated for the Hubbard model on the square lattice in [14, 15], consideration of the self-energy effects is very important. The normal self-energy component determines a reduction of the QP weight which can be quite large. The anomalous (pair) self-energy component strongly enhances the superconduction pairing induced by the kinematical interaction of electrons with spin-fluctuations. This interaction being proportional to the hopping parameter t [see equation (3.36)] is an order of magnitude larger than the exchange interaction J considered in the Anderson’s theory of the resonating valence bonds [24, 25]. We are planning to take into account the self-energy effects in a subsequent publication. Acknowledgements The author would like to thank A.A. Vladimirov and D. Ihle for valuable discussions. 33706-10 t-J model on the honeycomb lattice References 1. Plakida N.M., High-Temperature Cuprate Superconductors, Springer Series in Solid-State Sciences, Vol. 166, Springer-Verlag, Berlin, 2010. 2. Hubbard J., Proc. R. Soc. London, Ser. A, 1965, 285, 542, doi:10.1098/rspa.1965.0124. 3. 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З допомогою технiки проекцiйних операторiв Морi отримано рiвняння Дайсона для нормальних i аномальних функцiй Ґрiна для двозонної моделi в термiнах операторiв Хаббарда. Власно- енергетична частина розрахована у самоузгодженому борнiвському наближеннi для розсiяння електро- нiв на спiнових i зарядових флуктуацiях iндукованих кiнематичною взаємодiєю для операторiв Хаббарда. Обговорюється можливiсть надпровiдного спарювання через антиферомагнiтний обмiн та спiновi флук- туацiї. Ключовi слова: сильнi електроннi кореляцiї, t-J модель, стiльникова ґратка, надпровiднiсть 33706-11 https://doi.org/10.1098/rspa.1965.0124 https://doi.org/10.1142/S0217979288000524 https://doi.org/10.1142/S0217984988000771 https://doi.org/10.3367/UFNr.0071.196005c.0071 https://doi.org/10.1070/PU1960v003n03ABEH003275 https://doi.org/10.1016/0921-4534(89)90456-5 https://doi.org/10.1016/0921-4534(91)91576-P https://doi.org/10.1143/PTP.34.399 https://doi.org/10.1103/PhysRevB.59.11949 https://doi.org/10.1140/epjb/e2013-31157-6 https://doi.org/10.1134/S1063776114080123 https://doi.org/10.7868/S0044451014090235 https://doi.org/10.1103/RevModPhys.81.109 https://doi.org/10.1103/RevModPhys.84.1067 https://doi.org/10.1103/PhysRevLett.100.146404 https://doi.org/10.1103/PhysRevB.94.115105 https://doi.org/10.1103/PhysRevB.88.155112 https://doi.org/10.1140/epjb/e2017-70720-9 https://doi.org/10.1140/epjb/e2018-80712-x https://doi.org/10.1103/PhysRevLett.100.246808 https://doi.org/10.1126/science.235.4793.1196 Introduction The t-J model Dyson equation Mean-field approximation Self-energy operator Results Electronic spectrum Gap equation Conclusions

Electronic spectrum and superconductivity in the t-J model on the honeycomb lattice

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