Electron-hole asymmetry of quantum collective excitations in high- Tccopper oxides Maciej Fidrysiak1

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Electron-hole asymmetry of quantum collective excitations in high-Tccopper oxides

Maciej Fidrysiak1, ∗

1Institute of Theoretical Physics, Jagiellonian University, ul. Łojasiewicza 11, 30-348 Kraków, Poland

We carry out a systematic study of collective spin- and charge excitations for the canonical single-

band Hubbard, t-J-U, and t-Jmodels of high-temperature copper-oxide superconductors, both on

electron- and hole-doped side of the phase diagram. Recently developed variational wave function

approach, combined with the expansion in inverse number of fermionic ﬂavors, is employed. All

three models exhibit a substantial electron-hole asymmetry of magnetic excitations, with a robust

paramagnon emerging for hole-doping, in agreement with available resonant inelastic x-ray scattering

data for the cuprates. The t-Jmodel yields additional high-energy peak in the magnetic spectrum

that is not unambiguously identiﬁed in spectroscopy. For all considered Hamiltonians, the dynamical

charge susceptibility contains a coherent mode for both hole- and electron doping, with overall

bandwidth renormalization controlled by the on-site Coulomb repulsion. Away from the strong-

coupling limit, the antiferromagnetic ordering tendency is more pronounced on electron-doped side

of the phase diagram.

I. INTRODUCTION

High temperature superconductivity in layered copper

oxides may be induced by either hole (h) or electron (e)

doping of the parent antiferromagnetic (AF) insulating

state [1]. Nonetheless, the equilibrium phase diagrams of

e- and h-doped cuprates exhibit substantial asymmetry,

whose one of the most evident manifestations is broader

regime of the AF-phase appearance on the e-doped side

[2]. Thanks to developments in spectroscopic techniques,

particularly resonant inelastic x-ray scattering (RIXS),

this static picture has been recently supplemented with

a detailed account of non-equilibrium properties, includ-

ing collective excitation spectra [3–32]. The low-energy

dynamics of parent AF state is dominated by spin-waves

of localized spins, evolving with doping into broadened

magnetic excitations of the paramagnetic state (param-

agnons). Counter-intuitively, the paramagnons appear

to be more robust on the h-doped side, where AF phase

is rapidly suppressed away from half-ﬁlling. RIXS has

been also utilized to probe collective charge (plasmon)

excitations that are observed in both hole- [27–29] and

electron-doped [30–32] cuprates. Since magnetic and

charge ﬂuctuations have been considered as candidates

for the pairing mechanism in high-Tcsuperconductors

(see, e.g., [3, 33–35]), identiﬁcation of a microscopic ori-

gin of their distinct degree of e-h asymmetry is now in

demand.

Parent compounds of high-Tccopper-oxides are not

canonical Mott insulating systems, since they may be

regarded as charge transfer insulators within the Zaanen-

Sawatzky-Allen scheme [36]. In eﬀect, the e-h asymme-

try manifests itself directly at the atomic-orbital level.

Nominally, the doped electrons position on the copper

3dx2−y2states, while holes locate predominantly on 2pσ

oxygen orbitals. Yet, there is a systematic variation of d-

p-electron redistribution between diﬀerent cuprate fam-

∗maciej.ﬁdrysiak@uj.edu.pl

ilies [37]. Whereas minimally the d-pmodel of CuO2

plane is needed to address those orbital-selective eﬀects

[38–41], doping dependence of the principal experimen-

tal equilibrium properties and their e-h asymmetry is re-

produced already within the eﬀective single-band picture

[42–45]. The latter model is also appropriate for stud-

ies of the correlated state dynamics following interaction

quench [46, 47]. The single-band Hamiltonians remain

thus a common starting point for describing the physics

of high-Tccuprates and related materials.

Here we carry out a study of the e-h asymmetry of col-

lective magnetic and charge excitations for the canonical

Hubbard, t-J-U, and t-Jmodels of high-Tccopper-oxide

superconductors, which allows us to single out the mi-

croscopic parameters controlling relevant aspects of their

many-particle dynamics. The recently introduced con-

cept of eﬀective exchange interaction [48] is invoked to

relate all three Hamiltonians in proper limits so that

their collective excitations can be directly compared and

analyzed on equal footing. In order to account for the

eﬀects of strong electronic correlations, we supplement

Variational Wave Function (VWF) approach with the

expansion in inverse number of fermionic ﬂavors (1/Nf),

which results in VWF+1/Nfscheme [49]. The latter has

been demonstrated to be eﬀective in semi-quantitative

analysis of observed spin- and charge dynamics of h-

doped cuprates, and to compare favorably with other

computational techniques, including determinant quan-

tum Monte-Carlo [1, 48, 50, 51].

For the cases of Hubbard and t-J-Umodels up to mod-

erately large value of on-site interaction, we ﬁnd that in-

tense dispersive peak in magnetic spectrum persists along

the anti-nodal (Γ-X) Brillouin-zone direction down to

heavily h-overdoped regime, whereas the paramagnons

along the nodal (Γ-M) line are rapidly suppressed. On

the other hand, spin excitations are shifted to larger

energies with e-doping and become less coherent away

from half-ﬁlling. These ﬁndings indicate a substantial

e-h asymmetry at the dynamical level and are in agree-

ment with available RIXS data for the cuprates. More-

over, we ﬁnd that the magnetic spectra calculated in the

arXiv:2210.16109v3 [cond-mat.str-el] 13 Mar 2024

strong-coupling limit (t-Jmodel) diﬀer signiﬁcantly from

those obtained for the Hubbard and t-J-UHamiltonians,

and exhibit characteristics that are not observed exper-

imentally. This might suggest that the t-Jmodel over-

estimates local electronic correlations, as has been also

noted within former theoretical survey of equilibrium and

single-particle quantities [1].

For completeness, we also address the doping evolution

of collective charge excitations that are demonstrated to

persist on both h- and e-doped sides of the phase di-

agram. The dynamical charge response unambiguously

separates into the incoherent continuum part and a co-

herent charge mode. With increasing on-site Coulomb

repulsion, U, the collective mode energy undergoes a sys-

tematic downward renormalization. Thus, the electronic

band-narrowing eﬀects are instrumental for the charge

sector, whereas magnetic part remains primarily sensi-

tive to eﬀective exchange interaction, Jeﬀ . Remarkably,

within the t-J-Umodel, the parameters Uand Jeﬀ re-

main independent and may be simultaneously tuned to

match experiment. This is not possible within the Hub-

bard and t-Jmodels. Such ﬂexibility is needed to account

for non-trivial superexchange pathways via oxygen in a

charge-transfer insulator that alter the correspondence

between kinetic exchange and U[42].

Finally, we carry out analysis of the leading ordering

instabilities of the paramagnetic state both against spin-

and charge ﬂuctuations. For the Hubbard and t-J-U

models, we ﬁnd that the static magnetic susceptibility

at the MBrillouin-zone point is enhanced for e-doping

relative to the h-doped side, implying stronger tendency

towards antiferromagnetism. This reﬂects the experi-

mentally observed e-h asymmetry of the equilibrium AF

order.

II. MODEL AND METHOD

We employ a generic square-lattice t-J-Umodel of the

copper-oxygen plane, given by the Hamiltonian

H=X

i̸=j

tij ˆa†

iσ ˆajσ +JX

⟨i,j⟩

Si·ˆ

Sj+UX

ˆni↑ˆni↓,(1)

where ˆaiσ (ˆa†

iσ) are annihilation (creation) operators of

spin-σelectrons on site i,ˆniσ ≡ˆa†

iσ ˆaiσ, and ˆ

Si≡

(ˆ

i,ˆ

i)denotes local spin operator. Out of the

hopping integrals, tij , we retain only those connecting

nearest- and next-nearest neighbors, t≡ −0.35 eV and

t′≡0.25|t|, respectively. It should be remarked that,

within the t-J-Umodel, the degree of e-h asymmetry is

controlled exclusively by the magnitude of next-nearest

neighbor hopping, t′. Up to a trivial energy and chemical

potential shift, the remaining terms remain unchanged

after application of the e-h transformation in the usual

form

a†

iσ →aiσ ·(−1)i, aiσ →a†

iσ ·(−1)i,(2)

with (−1)ibeing phase factor alternating between neigh-

boring sites. The electronic interactions are governed by

the magnitude of on-site Coulomb repulsion, U, and by

nearest-neighbor AF exchange integral, J.

The t-J-UHamiltonian (1) may be regarded as a gen-

eralization of the Hubbard- and t-Jmodels, and encom-

passes both of them as particular cases. The Hubbard

model is obtained for U > 0and J= 0, whereas the t-J

model for U=∞and J > 0. In order to analyze all

three Hamiltonians on equal footing, we refer to the con-

cept of eﬀective exchange interaction Jeﬀ ≡J+4t2

Uthat

combines direct Heisenberg-type interaction with second-

order kinetic exchange [48]. Hereafter its value is set to

Jeﬀ ≡2

3|t|which has been used previously to study col-

lective dynamics in the paramagnetic state of h-doped

cuprates [51]. The selected representative parameter val-

ues are: U= 6|t|,J= 0 (Hubbard model), U= 12|t|,

J=1

3|t|(t-J-Umodel), and U=∞,J=2

3|t|(t-J

model).

We employ VWF+1/Nfapproach in the local-

diagrammatic variant (LDf+ 1/Nfin the notation of

Ref. [49]) that allows us to study collective excitations

around the strongly-correlated ground state. The saddle-

point reference point is constructed based on the varia-

tional wave function

|Ψvar⟩ ≡ ˆ

P|Ψ0⟩

|| ˆ

P|Ψ0⟩||,(3)

where |Ψ0⟩represents a Slater determinant, and ˆ

Pis

an operator introducing correlations into the trial state

(the so-called correlator). The denominator in Eq. (3) is

needed for normalization, since ˆ

Pis not unitary in gen-

eral. We take ˆ

P≡Qiˆ

Piwith

Pi≡λ0

i|0⟩ii⟨0|+X

σσ′

λσσ′

i|σ⟩ii⟨σ′|+λd

i|d⟩ii⟨d|,(4)

where states |0⟩i,|↑⟩i,|↓⟩i,|d⟩i≡ |↑↓⟩ispan the local

Hilbert space for the lattice site i, and σ, σ′=↑,↓enumer-

ate spin conﬁgurations. At zero temperature, both the

variational parameters λ=Si{λ0

i, λ↑↑

i, λ↑↓

i, λ↓↑

i, λ↓↓

i, λd

and wave function |Ψ0⟩are determined by optimization

of the energy functional Evar(|Ψ0⟩,λ)≡ ⟨Ψvar|ˆ

H|Ψvar⟩,

subjected to the requirement of ﬁxed total particle num-

ber and other constraints. Here we actually employ a

thermal generalization of this variational scheme based

on free energy functional [49], with temperature set to

kBT= 0.42|t|in order to stay clear of density-wave

orders. Stability of the paramagnetic state for all con-

sidered parameter sets is demonstrated in Appendix A,

where we also identify leading ordering tendencies. It

should be remarked that the transformation (2) preserves

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Electron-holeasymmetryofquantumcollectiveexcitationsinhigh-TccopperoxidesMaciejFidrysiak1,∗1InstituteofTheoreticalPhysics,JagiellonianUniversity,ul.Łojasiewicza11,30-348Kraków,PolandWecarryoutasystematicstudyofcollectivespin-andchargeexcitationsforthecanonicalsingle-bandHubbard,t-J-U,andt-Jmodelsofh...

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Electron-hole asymmetry of quantum collective excitations in high- Tccopper oxides Maciej Fidrysiak1

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