Quasiparticle Energy Dispersion and Shadow Peaks in a Doped 𝑺⁒𝑢⁒(πŸ“)𝑺𝑢5\bbox{SO(5)} Symmetric Ladder

Seung-Pyo Hong and Sung-Ho Suck Salk Department of Physics, Pohang University of Science and Technology, Pohang 790-784, Korea

Mean field and exact diagonalization studies on the quasiparticle excitation of an S⁒O⁒(5)𝑆𝑂5SO(5) symmetric two-leg ladder system are reported. It is shown that the energy gap in the quasiparticle excitation is caused by the formation of rung singlet states. We find that shadow peaks can occur above the Fermi surface with antiferromagnetic electron correlations involving only rungs in the spin ladder.

PACS numbers: 74.25.Jb, 74.25.-q, 71.10.-w, 71.27.+a

The cuprate materials of high critical superconducting temperatures exhibit antiferromagnetism near half-filling and superconductivity away from half filling. Zhang (1); (2)suggested that the two phenomena are manifestations of the same thing and can be explained in a unified framework based on S⁒O⁒(5)𝑆𝑂5SO(5) symmetry. Recently Scalapino et al. (3)presented a simple S⁒O⁒(5)𝑆𝑂5SO(5) model on a two-leg ladder system which can be served as a toy model to investigate various physics involving the S⁒O⁒(5)𝑆𝑂5SO(5) symmetry. In this paper we discuss quasiparticle excitations for the S⁒O⁒(5)𝑆𝑂5SO(5) two-leg spin ladder from both analytic and numerical results. We derive a mean field quasiparticle energy dispersion relation. For a through comparison one-particle spectral function is calculated by applying a Lanczos exact diagonalization method to a ladder of 2Γ—6262\times 6 sites. We find that the energy gap in the quasiparticle excitations is largely contributed by the formation of rung singlet states. We also find the appearance of shadow peaks with Heisenberg interaction along rungs alone.

The S⁒O⁒(5)𝑆𝑂5SO(5) symmetric two-leg ladder Hamiltonian is given by (3)

H𝐻\displaystyle H =\displaystyle= -tβˆ₯β’βˆ‘i⁒λ⁒σ(ci⁒λ⁒σ†⁒ci+1⁒λ⁒σ+H.c.)-tβŸ‚β’βˆ‘i⁒σ(ci⁒1⁒σ†⁒ci⁒2⁒σ+H.c.)subscript𝑑parallel-tosubscriptπ‘–πœ†πœŽsuperscriptsubscriptπ‘π‘–πœ†πœŽβ€ subscript𝑐𝑖1πœ†πœŽH.c.subscript𝑑perpendicular-tosubscriptπ‘–πœŽsuperscriptsubscript𝑐𝑖1πœŽβ€ subscript𝑐𝑖2𝜎H.c.\displaystyle-t_{\parallel}\sum_{i\lambda\sigma}(c_{i\lambda\sigma}^{\dagger}c% _{i+1\lambda\sigma}+\mbox{H.c.})-t_{\perp}\sum_{i\sigma}(c_{i1\sigma}^{\dagger% }c_{i2\sigma}+\mbox{H.c.}) (1)
+Uβ’βˆ‘i⁒λ(ni⁒λ↑-12)⁒(ni⁒λ↓-12)+Vβ’βˆ‘i(ni⁒1-1)⁒(ni⁒2-1)π‘ˆsubscriptπ‘–πœ†subscriptπ‘›β†‘π‘–πœ†absent12subscriptπ‘›β†“π‘–πœ†absent12𝑉subscript𝑖subscript𝑛𝑖11subscript𝑛𝑖21\displaystyle+U\sum_{i\lambda}\left(n_{i\lambda\uparrow}-\frac{1}{2}\right)% \left(n_{i\lambda\downarrow}-\frac{1}{2}\right)+V\sum_{i}(n_{i1}-1)(n_{i2}-1)
+Jβ’βˆ‘i𝐒i⁒1⋅𝐒i⁒2-ΞΌβ’βˆ‘i⁒λ⁒σni⁒λ⁒σ𝐽subscript𝑖⋅subscript𝐒𝑖1subscript𝐒𝑖2πœ‡subscriptπ‘–πœ†πœŽsubscriptπ‘›π‘–πœ†πœŽ\displaystyle+J\sum_{i}{\bf S}_{i1}\cdot{\bf S}_{i2}-\mu\sum_{i\lambda\sigma}n% _{i\lambda\sigma}

Here ci⁒λ⁒σ†superscriptsubscriptπ‘π‘–πœ†πœŽβ€ c_{i\lambda\sigma}^{\dagger} creates an electron with spin ΟƒπœŽ\sigma on the i𝑖i -th rung of the Ξ»πœ†\lambda -th leg with i=1,…,L𝑖1…𝐿i=1,\ldots,L and Ξ»=1,2πœ†12\lambda=1,2 . n𝑛n is the number operator, ni⁒λ⁒σ=ci⁒λ⁒σ†⁒ci⁒λ⁒σsubscriptπ‘›π‘–πœ†πœŽsuperscriptsubscriptπ‘π‘–πœ†πœŽβ€ subscriptπ‘π‘–πœ†πœŽn_{i\lambda\sigma}=c_{i\lambda\sigma}^{\dagger}c_{i\lambda\sigma} , ni⁒λ=βˆ‘Οƒni⁒λ⁒σsubscriptπ‘›π‘–πœ†subscript𝜎subscriptπ‘›π‘–πœ†πœŽn_{i\lambda}=\sum_{\sigma}n_{i\lambda\sigma} , and 𝐒𝐒{\bf S} is the spin operator, 𝐒i⁒λ=12⁒ciβ’Ξ»β’Ξ±β€ β’πˆΞ±β’Ξ²β’ci⁒λ⁒βsubscriptπ’π‘–πœ†12superscriptsubscriptπ‘π‘–πœ†π›Όβ€ subscriptπˆπ›Όπ›½subscriptπ‘π‘–πœ†π›½{\bf S}_{i\lambda}=\frac{1}{2}c_{i\lambda\alpha}^{\dagger}\mbox{\boldmath$% \sigma$}_{\alpha\beta}c_{i\lambda\beta} . tβˆ₯subscript𝑑parallel-tot_{\parallel} is the hopping integral in the leg direction; tβŸ‚subscript𝑑perpendicular-tot_{\perp} , the hopping integral in the rung direction; Uπ‘ˆU , the on-site Coulomb interaction; V𝑉V , the near-neighbor Coulomb interaction on a rung; J𝐽J , the Heisenberg exchange interaction on a rung; and ΞΌπœ‡\mu , the chemical potential. The Heisenberg interaction of the S⁒O⁒(5)𝑆𝑂5SO(5) ladder Hamiltonian allows both singly and doubly occupied sites, while that of the t𝑑t - J𝐽J ladder Hamiltonian allows only singly occupied sites. The constraint for the interaction strengths, J=4⁒(U+V)𝐽4π‘ˆπ‘‰J=4(U+V) in Eq. ( 1) above is required for the S⁒O⁒(5)𝑆𝑂5SO(5) symmetry. Uπ‘ˆU and V𝑉V can be repulsive or attractive potentials. In the present study we will consider only the case of repulsive potentials, Uβ‰₯0π‘ˆ0U\geq 0 and Vβ‰₯0𝑉0V\geq 0 , which is in the region of the E0subscript𝐸0E_{0} spin gap phase (represented by the tensor product of rung singlet states in the strong coupling limit) defined by Scalapino et al. (3)

The weak coupling (that is, U,Vβ‰ͺtβˆ₯≃tβŸ‚much-less-thanπ‘ˆπ‘‰subscript𝑑parallel-tosimilar-to-or-equalssubscript𝑑perpendicular-toU,V\ll t_{\parallel}\simeq t_{\perp} ) may allow linearization of the Coulomb repulsion terms

(ni⁒λ↑-12)⁒(ni⁒λ↓-12)subscriptπ‘›β†‘π‘–πœ†absent12subscriptπ‘›β†“π‘–πœ†absent12\displaystyle\left(n_{i\lambda\uparrow}-\frac{1}{2}\right)\left(n_{i\lambda% \downarrow}-\frac{1}{2}\right)
=\displaystyle= ⟨ni⁒λ↑-12⟩⁒(ni⁒λ↓-12)+⟨ni⁒λ↓-12⟩⁒(ni⁒λ↑-12)-⟨ni⁒λ↑-12⟩⁒⟨ni⁒λ↓-12⟩delimited-⟨⟩subscriptπ‘›β†‘π‘–πœ†absent12subscriptπ‘›β†“π‘–πœ†absent12delimited-⟨⟩subscriptπ‘›β†“π‘–πœ†absent12subscriptπ‘›β†‘π‘–πœ†absent12delimited-⟨⟩subscriptπ‘›β†‘π‘–πœ†absent12delimited-⟨⟩subscriptπ‘›β†“π‘–πœ†absent12\displaystyle\left\langle n_{i\lambda\uparrow}-\frac{1}{2}\right\rangle\left(n% _{i\lambda\downarrow}-\frac{1}{2}\right)+\left\langle n_{i\lambda\downarrow}-% \frac{1}{2}\right\rangle\left(n_{i\lambda\uparrow}-\frac{1}{2}\right)-\left% \langle n_{i\lambda\uparrow}-\frac{1}{2}\right\rangle\left\langle n_{i\lambda% \downarrow}-\frac{1}{2}\right\rangle


=\displaystyle= ⟨ni⁒1-1⟩⁒(ni⁒2-1)+⟨ni⁒2-1⟩⁒(ni⁒1-1)-⟨ni⁒1-1⟩⁒⟨ni⁒2-1⟩delimited-⟨⟩subscript𝑛𝑖11subscript𝑛𝑖21delimited-⟨⟩subscript𝑛𝑖21subscript𝑛𝑖11delimited-⟨⟩subscript𝑛𝑖11delimited-⟨⟩subscript𝑛𝑖21\displaystyle\langle n_{i1}-1\rangle(n_{i2}-1)+\langle n_{i2}-1\rangle(n_{i1}-% 1)-\langle n_{i1}-1\rangle\langle n_{i2}-1\rangle

Taking into account ⟨niβ’Ξ»β†‘βŸ©=⟨niβ’Ξ»β†“βŸ©=1-Ξ΄2delimited-⟨⟩subscriptπ‘›β†‘π‘–πœ†absentdelimited-⟨⟩subscriptπ‘›β†“π‘–πœ†absent1𝛿2\langle n_{i\lambda\uparrow}\rangle=\langle n_{i\lambda\downarrow}\rangle=% \frac{1-\delta}{2} with δ𝛿\delta , the doping rate, which neglects hole density fluctuations, we write the Coulomb repulsions (ni⁒λ↑-12)⁒(ni⁒λ↓-12)=-Ξ΄2⁒ni⁒λ+Ξ΄2-Ξ΄24subscriptπ‘›β†‘π‘–πœ†absent12subscriptπ‘›β†“π‘–πœ†absent12𝛿2subscriptπ‘›π‘–πœ†π›Ώ2superscript𝛿24(n_{i\lambda\uparrow}-\frac{1}{2})(n_{i\lambda\downarrow}-\frac{1}{2})=-\frac{% \delta}{2}n_{i\lambda}+\frac{\delta}{2}-\frac{\delta^{2}}{4} and (ni⁒1-1)⁒(ni⁒2-1)=-δ⁒(ni⁒1+ni⁒2)+2⁒δ-Ξ΄2subscript𝑛𝑖11subscript𝑛𝑖21𝛿subscript𝑛𝑖1subscript𝑛𝑖22𝛿superscript𝛿2(n_{i1}-1)(n_{i2}-1)=-\delta(n_{i1}+n_{i2})+2\delta-\delta^{2} , and add them to the chemical potential term in Eq. ( 1). The Heisenberg interaction can be written (4)as

𝐒i⁒1⋅𝐒i⁒2β‹…subscript𝐒𝑖1subscript𝐒𝑖2\displaystyle{\bf S}_{i1}\cdot{\bf S}_{i2} =\displaystyle= -38⁒[Ο‡i⁒12βˆ—β’(ci⁒1↑†⁒ci⁒2↑+ci⁒1↓†⁒ci⁒2↓)+H.c.]+38⁒ni⁒138delimited-[]superscriptsubscriptπœ’π‘–12βˆ—superscriptsubscript𝑐↑𝑖1absent†subscript𝑐↑𝑖2absentsuperscriptsubscript𝑐↓𝑖1absent†subscript𝑐↓𝑖2absentH.c.38subscript𝑛𝑖1\displaystyle-\frac{3}{8}[\chi_{i12}^{\ast}(c_{i1\uparrow}^{\dagger}c_{i2% \uparrow}+c_{i1\downarrow}^{\dagger}c_{i2\downarrow})+\mbox{H.c.}]+\frac{3}{8}% n_{i1}
-38⁒[Ξ”i⁒12βˆ—β’(ci⁒1↑⁒ci⁒2↓-ci⁒1↓⁒ci⁒2↑)+H.c.]38delimited-[]superscriptsubscriptΔ𝑖12βˆ—subscript𝑐↑𝑖1absentsubscript𝑐↓𝑖2absentsubscript𝑐↓𝑖1absentsubscript𝑐↑𝑖2absentH.c.\displaystyle-\frac{3}{8}[\Delta_{i12}^{\ast}(c_{i1\uparrow}c_{i2\downarrow}-c% _{i1\downarrow}c_{i2\uparrow})+\mbox{H.c.}]

where the hopping order parameter is Ο‡i⁒12=⟨ci⁒1↑†⁒ci⁒2↑+ci⁒1↓†⁒ci⁒2β†“βŸ©subscriptπœ’π‘–12delimited-⟨⟩superscriptsubscript𝑐↑𝑖1absent†subscript𝑐↑𝑖2absentsuperscriptsubscript𝑐↓𝑖1absent†subscript𝑐↓𝑖2absent\chi_{i12}=\langle c_{i1\uparrow}^{\dagger}c_{i2\uparrow}+c_{i1\downarrow}^{% \dagger}c_{i2\downarrow}\rangle and the singlet pair order parameter, Ξ”i⁒12=⟨ci⁒1↑⁒ci⁒2↓-ci⁒1↓⁒ci⁒2β†‘βŸ©subscriptΔ𝑖12delimited-⟨⟩subscript𝑐↑𝑖1absentsubscript𝑐↓𝑖2absentsubscript𝑐↓𝑖1absentsubscript𝑐↑𝑖2absent\Delta_{i12}=\langle c_{i1\uparrow}c_{i2\downarrow}-c_{i1\downarrow}c_{i2% \uparrow}\rangle . We take Ο‡i⁒12=Ο‡subscriptπœ’π‘–12πœ’\chi_{i12}=\chi and Ξ”i⁒12=Ξ”subscriptΔ𝑖12Ξ”\Delta_{i12}=\Delta by neglecting spatial fluctuations of both the amplitude and the phase. The mean field Hamiltonian is then in momentum space,

H𝐻\displaystyle H =\displaystyle= βˆ‘k⁒λ⁒σ(-2⁒tβˆ₯⁒cos⁑kx-ΞΌ)⁒ck⁒λ⁒σ†⁒ck⁒λ⁒σ-tβŸ‚β’βˆ‘k⁒σ(ck⁒1⁒σ†⁒ck⁒2⁒σ+H.c.)subscriptπ‘˜πœ†πœŽ2subscript𝑑parallel-tosubscriptπ‘˜π‘₯πœ‡superscriptsubscriptπ‘π‘˜πœ†πœŽβ€ subscriptπ‘π‘˜πœ†πœŽsubscript𝑑perpendicular-tosubscriptπ‘˜πœŽsuperscriptsubscriptπ‘π‘˜1πœŽβ€ subscriptπ‘π‘˜2𝜎H.c.\displaystyle\sum_{k\lambda\sigma}(-2t_{\parallel}\cos k_{x}-\mu)c_{k\lambda% \sigma}^{\dagger}c_{k\lambda\sigma}-t_{\perp}\sum_{k\sigma}(c_{k1\sigma}^{% \dagger}c_{k2\sigma}+\mbox{H.c.}) (2)
-3⁒J8β’βˆ‘k[Ο‡βˆ—β’(ck⁒1↑†⁒ck⁒2↑+ck⁒1↓†⁒ck⁒2↓)+Ξ”βˆ—β’(ck⁒1↑⁒c-k⁒2↓-ck⁒1↓⁒c-k⁒2↑)+H.c.]3𝐽8subscriptπ‘˜delimited-[]superscriptπœ’βˆ—superscriptsubscriptπ‘β†‘π‘˜1absent†subscriptπ‘β†‘π‘˜2absentsuperscriptsubscriptπ‘β†“π‘˜1absent†subscriptπ‘β†“π‘˜2absentsuperscriptΞ”βˆ—subscriptπ‘β†‘π‘˜1absentsubscriptπ‘β†“π‘˜2absentsubscriptπ‘β†“π‘˜1absentsubscriptπ‘β†‘π‘˜2absentH.c.\displaystyle-\frac{3J}{8}\sum_{k}[\chi^{\ast}(c_{k1\uparrow}^{\dagger}c_{k2% \uparrow}+c_{k1\downarrow}^{\dagger}c_{k2\downarrow})+\Delta^{\ast}(c_{k1% \uparrow}c_{-k2\downarrow}-c_{k1\downarrow}c_{-k2\uparrow})+\mbox{H.c.}]

The quasiparticle energy dispersion is readily obtained from Eq. ( 2) above,

Ek=Β±[(-2⁒tβˆ₯⁒cos⁑kx-ΞΌ)Β±(tβŸ‚+3⁒J⁒χ8)]2+(3⁒J⁒Δ8)2subscriptπΈπ‘˜plus-or-minussuperscriptdelimited-[]plus-or-minus2subscript𝑑parallel-tosubscriptπ‘˜π‘₯πœ‡subscript𝑑perpendicular-to3π½πœ’82superscript3𝐽Δ82E_{k}=\pm\sqrt{\left[(-2t_{\parallel}\cos k_{x}-\mu)\pm\left(t_{\perp}+\frac{3% J\chi}{8}\right)\right]^{2}+\left(\frac{3J\Delta}{8}\right)^{2}} (3)

The mean field order parameters Ο‡πœ’\chi , ΔΔ\Delta , and the chemical potential ΞΌπœ‡\mu are obtained from self-consistent equations. For the time being we take the values of Ο‡πœ’\chi , ΔΔ\Delta , and ΞΌπœ‡\mu obtained from our Lanczos calculations on the 2Γ—6262\times 6 ladder with two doped holes. The quasiparticle energy dispersion is shown in Fig. 1; the bonding band (denoted as B) has its minimum energy at kx=0subscriptπ‘˜π‘₯0k_{x}=0 with the Fermi surface at kFB≃2⁒π3similar-to-or-equalssuperscriptsubscriptπ‘˜πΉπ΅2πœ‹3k_{F}^{B}\simeq\frac{2\pi}{3} , while the antibonding band (denoted as A) has its minimum energy at kx=0subscriptπ‘˜π‘₯0k_{x}=0 with the Fermi surface at kFA≃π3similar-to-or-equalssuperscriptsubscriptπ‘˜πΉπ΄πœ‹3k_{F}^{A}\simeq\frac{\pi}{3} . Splitting between the two bands, that is, the removal of degeneracy is caused by hopping (overlap) integral tβŸ‚subscript𝑑perpendicular-tot_{\perp} in the rung direction. The bonding band is pushed down with more occupied electrons, while the antibonding band is pushed up with less occupied electrons. We note that with kFB>kFAsuperscriptsubscriptπ‘˜πΉπ΅superscriptsubscriptπ‘˜πΉπ΄k_{F}^{B}>k_{F}^{A} the Luttinger sum rule is satisfied, that is, kFB+kFA=(1-Ξ΄)⁒πsuperscriptsubscriptπ‘˜πΉπ΅superscriptsubscriptπ‘˜πΉπ΄1π›Ώπœ‹k_{F}^{B}+k_{F}^{A}=(1-\delta)\pi . The dashed parts of the dispersion curves in Fig. 1represent the shadow bands in our two-leg ladder system which is similar to the ones discussed in two-dimensional planar systems (5); (6); (7). As shown in Eq. ( 3) the shadow band in Fig. 1is largely contributed by the Heisenberg interaction in the rung direction, while the shadow band in the two-dimensional planar systems (5); (6); (7)is caused by antiferromagnetic correlations whose strengths are equal in both the horizontal and vertical directions.

To examine the single particle excitations, we write in the case of two-hole doped system of 2Γ—L2𝐿2\times L sites (8),

Ae⁒(𝐀,Ο‰)=βˆ‘Ξ±|⟨Ψα2⁒L-1|c𝐀⁒σ†|Ξ¨02⁒L-2⟩|2⁒δ⁒(Ο‰-EΞ±2⁒L-1+E02⁒L-2+ΞΌ)subscriptπ΄π‘’π€πœ”subscript𝛼superscriptquantum-operator-productsuperscriptsubscriptΨ𝛼2𝐿1superscriptsubscriptπ‘π€πœŽβ€ superscriptsubscriptΞ¨02𝐿22π›Ώπœ”superscriptsubscript𝐸𝛼2𝐿1superscriptsubscript𝐸02𝐿2πœ‡A_{e}({\bf k},\omega)=\sum_{\alpha}\left|\langle\Psi_{\alpha}^{2L-1}|c_{{\bf k% }\sigma}^{\dagger}|\Psi_{0}^{2L-2}\rangle\right|^{2}\delta(~{}~{}\omega-E_{% \alpha}^{2L-1}+E_{0}^{2L-2}+\mu) 4.a

for the particle spectral function (Ο‰>0)πœ”0(\omega>0) and

Ah⁒(𝐀,Ο‰)=βˆ‘Ξ±|⟨Ψα2⁒L-3|c𝐀⁒σ|Ξ¨02⁒L-2⟩|2⁒δ⁒(-Ο‰-EΞ±2⁒L-3+E02⁒L-2-ΞΌ)subscriptπ΄β„Žπ€πœ”subscript𝛼superscriptquantum-operator-productsuperscriptsubscriptΨ𝛼2𝐿3subscriptπ‘π€πœŽsuperscriptsubscriptΞ¨02𝐿22π›Ώπœ”superscriptsubscript𝐸𝛼2𝐿3superscriptsubscript𝐸02𝐿2πœ‡A_{h}({\bf k},\omega)=\sum_{\alpha}\left|\langle\Psi_{\alpha}^{2L-3}|c_{{\bf k% }\sigma}|\Psi_{0}^{2L-2}\rangle\right|^{2}\delta(-\omega-E_{\alpha}^{2L-3}+E_{% 0}^{2L-2}-\mu) 4.b

for the hole spectral function (Ο‰<0)πœ”0(\omega<0) . Here |ΨαN⟩ketsuperscriptsubscriptΨ𝛼𝑁|\Psi_{\alpha}^{N}\rangle is the α𝛼\alpha -th eigenstate in the subspace of N𝑁N electrons with the eigenenergy EΞ±Nsuperscriptsubscript𝐸𝛼𝑁E_{\alpha}^{N} . The chemical potential is defined as ΞΌ=12⁒(E02⁒L-1-E02⁒L-3)πœ‡12superscriptsubscript𝐸02𝐿1superscriptsubscript𝐸02𝐿3\mu=\frac{1}{2}(E_{0}^{2L-1}-E_{0}^{2L-3}) . We calculate the spectral function by using both the Lanczos method and the continued fraction approach (9). The results are shown in Fig. 2for L=6𝐿6L=6 , tβˆ₯=tβŸ‚=1subscript𝑑parallel-tosubscript𝑑perpendicular-to1t_{\parallel}=t_{\perp}=1 and V=0𝑉0V=0 , as a function of Uπ‘ˆU and J𝐽J . The coupling strengths, Uπ‘ˆU and J𝐽J are related to each other and J𝐽J increases faster than Uπ‘ˆU due to the S⁒O⁒(5)𝑆𝑂5SO(5) constraint, J=4⁒(U+V)=4⁒U𝐽4π‘ˆπ‘‰4π‘ˆJ=4(U+V)=4U .

We now investigate the spectral functions by varying the values of Uπ‘ˆU (and consequently J𝐽J ). First the computed free particle spectral functions for the case of U=J=0π‘ˆπ½0U=J=0 are shown in Fig. 2(a). The Fermi momentum of the bonding (ky=0)subscriptπ‘˜π‘¦0(k_{y}=0) band is kFB=2⁒π3superscriptsubscriptπ‘˜πΉπ΅2πœ‹3k_{F}^{B}=\frac{2\pi}{3} , and that of the antibonding (ky=Ο€)subscriptπ‘˜π‘¦πœ‹(k_{y}=\pi) band is kFA=Ο€3superscriptsubscriptπ‘˜πΉπ΄πœ‹3k_{F}^{A}=\frac{\pi}{3} . Below the Fermi surface, we observe sharp and well-defined quasiparticle peaks for both the bonding and the antibonding bands. All the single particle states below the Fermi surface are occupied, and there exist no spectral peaks above the Fermi surface. The computed spectral functions shown in Fig. 2(a) are seen to agree well with the free particle energy dispersion, Ξ΅k=-2⁒tβˆ₯⁒cos⁑kxΒ±tβŸ‚subscriptπœ€π‘˜plus-or-minus2subscript𝑑parallel-tosubscriptπ‘˜π‘₯subscript𝑑perpendicular-to\varepsilon_{k}=-2t_{\parallel}\cos k_{x}\pm t_{\perp} in Eq. ( 3).

The spectral functions for the case of small coupling strengths, U=0.1π‘ˆ0.1U=0.1 and J=0.4𝐽0.4J=0.4 are displayed in Fig. 2(b). The predicted Fermi momenta are found at kFB≃2⁒π3similar-to-or-equalssuperscriptsubscriptπ‘˜πΉπ΅2πœ‹3k_{F}^{B}\simeq\frac{2\pi}{3} and kFA≃π3similar-to-or-equalssuperscriptsubscriptπ‘˜πΉπ΄πœ‹3k_{F}^{A}\simeq\frac{\pi}{3} in agreement with the mean field result given by Eq. ( 3). The calculated positions of the quasiparticle peaks below the Fermi surface are also in good agreement with the analytic mean field result of Eq. ( 3). Although the interaction strengths are small, interestingly enough there appear spectral peaks above the Fermi momentum (surface). Such peaks are the shadow peaks which were also observed by others (10). The peak in the bonding band with momentum (Ο€,0)πœ‹0(\pi,0) above the Fermi surface is a shadow of the peak belonging to the antibonding band with momentum (0,Ο€)0πœ‹(0,\pi) which is below the Fermi surface. The peak with momentum (2⁒π3,Ο€)2πœ‹3πœ‹(\frac{2\pi}{3},\pi) is a shadow of the peak with momentum (-Ο€3,0)πœ‹30(-\frac{\pi}{3},0) (degenerate at (Ο€3,0)πœ‹30(\frac{\pi}{3},0) ), and the peak with momentum (Ο€,Ο€)πœ‹πœ‹(\pi,\pi) is a shadow of the peak with momentum (0,0)00(0,0) . Haas and Dagotto (10)reported the presence of shadow peaks in t𝑑t - J𝐽J ladder systems and concluded that short-range antiferromagnetic correlations are responsible for the shadow peaks in the spin ladder system.

In order to investigate which of the two parameters Uπ‘ˆU and J𝐽J is more responsible for yielding the shadow peaks, we first set J=0𝐽0J=0 and calculate spectral functions for several cases of Uπ‘ˆU as shown in Fig. 3. We note that the stronger the on-site Coulomb repulsion, the broader the bonding orbital becomes, which is more evident for the orbitals, particularly at lower values of kxsubscriptπ‘˜π‘₯k_{x} (substantially below the Fermi surface), for example, at k=(0,0)π‘˜00k=(0,0) and (Ο€3,0)πœ‹30(\frac{\pi}{3},0) as shown in Fig. 3. The overall structure of the bonding and the antibonding bands does not change much with increasing Uπ‘ˆU . We find that the shadow peaks can be generated even with a small Coulomb repulsion U=0.1π‘ˆ0.1U=0.1 as shown in Fig. 3(b). While Haas and Dagotto’s results (10)are based on the strong coupling limit U≫tmuch-greater-thanπ‘ˆπ‘‘U\gg t due to the use of the t𝑑t - J𝐽J Hamiltonian, we discover the shadow peaks in both the weak and the strong coupling limits. Second order hopping processes generates an effective interaction of strength ∼t2/Usimilar-toabsentsuperscript𝑑2π‘ˆ\sim t^{2}/U between the electrons on nearest-neighbor sites. Since the hopping integral along the rung ( tβŸ‚subscript𝑑perpendicular-tot_{\perp} ) and that along the chain ( tβˆ₯subscript𝑑parallel-tot_{\parallel} ) are the same, it is hard to determine from their study which direction of electron correlations is more important for the shadow peaks.

In order to thoroughly verify the validity of shadow peaks caused essentially by the antiferromagnetic correlation between the two electrons in the rung, namely, by the spin singlet state along the rung direction, we computed the spectral functions for the case of U=V=0π‘ˆπ‘‰0U=V=0 , JβŸ‚β‰‘Jβ‰ 0subscript𝐽perpendicular-to𝐽0J_{\perp}\equiv J\neq 0 and Jβˆ₯=0subscript𝐽parallel-to0J_{\parallel}=0 as shown in Fig. 4. We find that the shadow peaks appear above the Fermi surface with the Heisenberg interaction along rungs alone as displayed in Fig. 4(b). Thus the singlet bonding on the rungs is responsible for the presence of the shadow peaks.

Now we reexamine the S⁒O⁒(5)𝑆𝑂5SO(5) symmetric cases of J=4⁒U𝐽4π‘ˆJ=4U . The spectral functions for different values of Uπ‘ˆU and J𝐽J are shown in Figs. 2(c),(d),(e). As J𝐽J increases, the energies of the bonding and the antibonding orbitals decrease with increasing energy gap. It is to be reminded that due to the S⁒O⁒(5)𝑆𝑂5SO(5) constraint to define the relation between J𝐽J and Uπ‘ˆU , J𝐽J increases more rapidly (4 times faster) than Uπ‘ˆU . Thus it is of great interest to see how the spectral positions and energy gaps vary as a result of larger contribution by J𝐽J . As can be seen from the comparison of Figs. 2and 3, we note that the energy gap is largely contributed by J𝐽J , but not substantially so by Uπ‘ˆU . The dominance of spin singlet states on the rungs for the determination of the energy gap is manifest from this study. Indeed this observation is in good accordance with the mean field prediction; the energy gap is caused by the Heisenberg interaction J𝐽J as shown in Eq. ( 3).

The dispersion of the bonding band does not greatly change with increasing values of J𝐽J , in contrast to that of the antibonding band. As J𝐽J increases, the antibonding band is lowered more rapidly than the bonding band and finally becomes dispersionless (no dependence on kxsubscriptπ‘˜π‘₯k_{x} ) as shown in Fig. 2(e). The Heisenberg interaction J𝐽J , which is taken to be anisotropic due to JβŸ‚β‰ 0subscript𝐽perpendicular-to0J_{\perp}\neq 0 and Jβˆ₯=0subscript𝐽parallel-to0J_{\parallel}=0 , obviously favors the formation of the local rung singlet. The rung singlets are clearly local because there exists no phase coherence between the singlets on the adjacent rungs, which, in turn, causes the appearance of the dispersionless antibonding band.

We have investigated quasiparticle excitations in the S⁒O⁒(5)𝑆𝑂5SO(5) symmetric two-leg ladder system recently proposed by Scalapino et al. (3)The quasiparticle energy dispersion is discussed in both the mean field approach and the exact diagonalization study. The dispersion consists of two branches, the bonding band with Fermi momentum kFB≃2⁒π3similar-to-or-equalssuperscriptsubscriptπ‘˜πΉπ΅2πœ‹3k_{F}^{B}\simeq\frac{2\pi}{3} and the antibonding band with Fermi momentum kFA≃π3similar-to-or-equalssuperscriptsubscriptπ‘˜πΉπ΄πœ‹3k_{F}^{A}\simeq\frac{\pi}{3} . The hole spectral functions calculated from the Lanczos exact diagonalization method were found to agree well with the mean field results. The formation of rung singlet states is responsible for the energy gap in the quasiparticle excitations. Finally we find that the presence of shadow peaks above the Fermi surface can essentially occur as a result of the singlet bonding on the rungs, indicating that the on-site Coulomb repulsion along chains or rungs may not be a prime cause for the formation of shadow peaks.

One of us (S.H.S.S) acknowledges the financial supports of Korean Ministry of Education (BSRI-97) and of the Center for Molecular Sciences at KAIST.


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  • FIG. 1.

    The quasiparticle energy dispersion obtained by using the mean field approach in the weak coupling limit. The parameters are chosen from our Lanczos calculations on a 2Γ—6262\times 6 ladder with two doped holes as tβˆ₯=tβŸ‚=1subscript𝑑parallel-tosubscript𝑑perpendicular-to1t_{\parallel}=t_{\perp}=1 , J=0.4𝐽0.4J=0.4 , Ο‡=0.47πœ’0.47\chi=0.47 , Ξ”=-0.15Ξ”0.15\Delta=-0.15 , ΞΌ=-0.099πœ‡0.099\mu=-0.099 . B denotes a bonding ( ky=0subscriptπ‘˜π‘¦0k_{y}=0 ) band and A, an antibonding ( ky=Ο€subscriptπ‘˜π‘¦πœ‹k_{y}=\pi ) band. The dashed parts of the dispersions denote shadow bands.

  • FIG. 2.

    Exact diagonalization results of the hole spectral function on a 2Γ—6262\times 6 ladder with two doped holes for tβˆ₯=tβŸ‚=1subscript𝑑parallel-tosubscript𝑑perpendicular-to1t_{\parallel}=t_{\perp}=1 , V=0𝑉0V=0 , and several values of Uπ‘ˆU and J𝐽J . The dashed horizontal line denotes the Fermi energy, and the dotted arrows indicate the shadow peaks. The δ𝛿\delta functions have been given a finite width of Ο΅=0.1italic-Ο΅0.1\epsilon=0.1 .

  • FIG. 3.

    Same as in Fig. 2 but for J=0𝐽0J=0 .

  • FIG. 4.

    Same as in Fig. 2 but for U=0π‘ˆ0U=0 .

FigureΒ 1:
FigureΒ 2:
FigureΒ 3:
FigureΒ 4: