We apply a diagrammatic expansion method around the atomic limit (U >> t) for
the U-t-t' Hubbard model at half filling and finite temperature by means of a
continued fraction representation of the one-particle Green's function. From
the analysis of the spectral function A(\vec{k},\omega) we find an energy
dispersion relation with a (cos k_x-cos k_y)^2 modulation of the energy gap in
the insulating phase. This anisotropy is compared with experimental ARPES
results on insulating cuprates.Comment: 4 pages Revtex, 6 embedded eps figures; Figures 5 and 6 were in error
  and have been replaced including the discussion of the figure

Brune, Ph.

Kampf, A. P.

English

arXiv

Brune, Philipp

Kampf, Arno P. (Prof.)

Online-Publikationserver Augsburg

Anisotropy of the energy gap in the insulating phase of the U-t-t' Hubbard model

OPUS Augsburg

                                                                                                              Anisotropy of the energy gap in the insulating phaseof the U-t-t′ Hubbard modelPh. Brunea and A.P. KampfInstitut für Physik, Theoretische Physik III, Elektronische Korrelationen und Magnetismus, Universität Augsburg,86135 Augsburg, Germany                                                 Abstract. We apply a diagrammatic expansion method around the atomic limit (U  t) for the U-t-t′Hubbard model at half filling and finite temperature by means of a continued fraction representation ofthe one-particle Green’s function. From the analysis of the spectral function A(k, ω) we find an energydispersion relation with a (cos kx − cos ky)2 modulation of the energy gap in the insulating phase. Thisanisotropy is compared with experimental ARPES results on insulating cuprates.PACS. 71.10.Fd Lattice fermion models (Hubbard model, etc.) – 71.27.+a Strongly correlated electronsystems; heavy fermions – 71.30.+h Metal-insulator transitions and other electronic transitionsDuring the last decade of research work on high tem-perature superconductors angular resolved photoemissionspectroscopy (ARPES) has played an important role inelucidating their electronic properties [1]. An understand-ing of the single particle properties in the normal stateis a prerequisite for a theory of a mechanism for super-conductivity as well as for transport properties. Over theyears ARPES data have continuously provided surprisesand new insights and thereby served as a guidance to theo-retical developments. The observation of the dx2−y2 -shapeof the energy gap, pseudogap structures in the metallicphase [2,3], strong anisotropies in the quasiparticle (qp)peak lineshapes and a possible partial destruction of theFermi surface [4], or the unusual frequency and temper-ature dependence of the qp peakwidth [5] are examplesfor intriguing information obtained from ARPES experi-ments.For the normal state an important goal is the descrip-tion for the evolution with doping from the antiferromag-netic (AF) and insulating parent compounds to the over-doped superconductors. This demands control over thespectral features of the Mott-Hubbard insulating state asa starting point. Yet, it proved to be difficult to repro-duce ARPES data for the single hole dispersion in theinsulating cuprate Sr2CuO2Cl2 [6] within t-J or Hubbardmodels. In particular, for momenta along the Brillouinzone (BZ) axis next-nearest neighbor (nnn) or even longerrange hopping amplitudes had to be introduced to achievea reasonable comparison to the measured spectra [7–15].Even more striking are recent results on AF Ca2CuO2Cl2242                            with the partition function Z and the action S = Skin +Satom whereSkin[γ∗, γ] = −∫ β0dτ∑i,j,σtij(γ∗iστγjστ + h.c.) , (2)Satom[γ∗, γ] =∫ β0dτ∑i,σγ∗iστ (∂                                                                                        243by the diagrammatic expansion to a Green’s function GJin a finite Jacobi continued fraction representation [24]GJ (z) =a0244                            (0,0) (π,π) (π,0) (0,0)−1.3−1.2−1.1−1.0−0.9−0.8−0.7E k [eV](π,0) (0,π)Fig. 4. A comparison with the ARPES dispersion forSr2CuO2Cl2 taken from reference [6] (black diamonds) andreference [17] (white circles). Error bars are shown at someselected experimental data points. The parameters in the the-ory were chosen as U = 10t, T = 0.2t, t′ = −0.45t, andt = 0.605 eV.results – is obtained naturally within the presented strong-coupling theory for the Hubbard model with nnn hoppingonly.The deviation of the theoretical result from the exper-imental data points along the BZ diagonal, i.e. the lackof symmetry with respect to kd – as realized in the AFstate – is due to the finite temperature. In fact, the rangeof applicability of the t/U expansion has a lower boundin temperature. (Following the arguments of Ref. [20] weestimate that our results are valid for T > 0.16t.) Whenthe magnetic correlation length is much larger than thehopping range as given by the order of the t/U expan-sion, an accurate description of the single particle Green’sfunction is no longer expected. The data were taken 100 Kabove the Néel temperature TN = 256 K of Sr2CuO2Cl2;at this temperature the magnetic correlation length is al-ready as large as 250 Å as measured by neutron scattering[30]. Nevertheless, the flatness of the dispersion along theBZ axis in the t/U expansion is a robust and temperatureinsensitive feature.In Figure 5 the calculated energy dispersion E(kd) −E(k) of the lowest energy peak and the experimental dataare plotted along the BZ path from kd to (π, 0) which isparameterized in the form |γd(k)|/2. Originally, the dataof reference [16] (black squares) were anticipated to implya dx2−y2-modulation of the energy gap which in the pa-rametrization of Figure 5 would translate into a straightline. The perfectly linear relation between E(kd) − E(k)and |γd(k)| along the chosen path would furthermore im-ply an unreasonable cusp-like feature of the dispersion onthe magnetic BZ boundary. Indeed, the more recent databy Ronning et al. [17] (white circles in Fig. 5) resolve thisproblem and rather provide evidence for a quadratic de-pendence of the energy gap on |γd(k)| in the vicinity ofkd. Given the experimental error bars of reference [6] in-cluded in Figure 5 and in the absence of an error estimatefor the recent data by Ronning et al. [17] our results are0.0 0.2 0.4 0.6 0.8 1.0|cos(kx)−cos(ky)|/2−0.10.00.10.20.30.40.5E (π/2,π/2)−E kFig. 5. Same data as in Figure 4 along the BZ path fromkd = (π/2, π/2) to (π, 0), but here E(kd) − E(k) is plot-ted as a function of |γd(k)|/2. In addition, ARPES data onCa2CuO2Cl2 (from Ref. [16]) are shown (black squares).0.0 0.2 0.4 0.6 0.8 1.0(cos(kx)−cos(ky))2/40.000.100.200.300.400.50E (π/2,π/2)−E k T=0.2tT=0.3tT=0.5tT=0.7tFig. 6. Dispersion along the path from kd = (π/2, π/2) to(π, 0) relative to its value at kd as a function of (cos kx −cos ky)2/4 for different temperatures with U = 10t and t′ =−0.4t.clearly compatible with experiment. With the same pa-rameter set used for the fit of the ARPES dispersion inFigure 4 the gap modulation amplitude ∼ 300 meV asmeasured in Ca2CuO2Cl2 [16] is reproduced as well.Figure 6 shows the temperature dependence of the gapmodulation. In fact, when plotted versus γ2d(k) it becomesevident that the gap approaches a perfect γ2d(k) momen-tum dependence at low temperatures. For t′ = 0 the mod-ulation vanishes in the expansion up to the order (t/U)4because the k dependence of all diagrams to this orderarises in the form coskx + cosky for t′ = 0 which vanisheson the BZ boundary for t′ = 0. If diagrams of order (t/U)6or higher were taken into account, a small modulation isexpected to appear also for t′ = 0.It is an interesting observation that the SDW mean-field solution of the U -t-t′ Hubbard model, where theenergy dispersion takes the form of two energy bands                                                                                        245E±(k) = −4t′ cos kx cos ky±√

OPUS - Augsburg University Publication Server

We apply a diagrammatic expansion method around the atomic limit 
($U \gg t$) for the U-t-t' Hubbard model at half filling and finite 
temperature by means of a continued fraction representation of the 
one-particle Green's function. From the analysis of the spectral function 
$A({\bf k},\omega)$ we find an energy dispersion relation with a 
$(\cos k_x-\cos k_y)^2$ modulation of the energy gap in the insulating phase. 
This anisotropy is compared with experimental ARPES results on insulating 
cuprates

Ph. Brune

A. P. Kampf

EDP Sciences OAI-PMH repository (1.2.0)

Anisotropy of the energy gap in the insulating phase 
of the U-t-t' Hubbard model

A.P. Kampf

Crossref

Anisotropy of the Energy Gap in the Insulating Phase of the U-t-t'
  Hubbard Model

Anisotropy of the Energy Gap in the Insulating Phase of the U-t-t' Hubbard Model

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