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Inelastic X-ray scattering in strongly correlated (Mott)

insulatorsT. P. Devereaux

With J. Freericks (Georgetown).

Work supported by NSERC and PREA.

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Quantum Critical PointsQuantum Critical Points

-one particle properties may be uncritical, two particle properties may not.

EXAMPLE:EXAMPLE:

(Anderson) metal-insulator transition

1/ , DOS – non-critical, - falls to zero at MIT.

Cuprates phase diagram

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Experimental data for the cuprates

• reduction of low-frequency spectral weight• increase in the charge transfer peak• isosbestic point at about 2100 cm-1.

Irwin et al, 1998.

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Common to other systems?FeSi – Kondo Insulator SmB6 – mixed valent

insulator

• transfer of spectral weight from low frequencies to high as T reduced.

• occurrence of “isosbestic point” (spectrum independent of T).

• qualitatively similar to B1g in underdoped cuprates.

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Low energy features.

F. Venturini et al, 2002.

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Shows a clear break in behavior at a doping pc ~ 0.22.

Indicates that the “hot” qps become incapable of carrying current.

-> unconventional quantum critical metal – insulator transition for p=pc.

Venturini et al, 2002.

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Inelastic X-ray scatteringM. Hasan et al, 2001 – Ca2 Cu O2 Cl2

• non-dispersive peak ~ 5.8 eV

• weak, dispersive peak ~ 2.5-4 eV

•which features are associated with excitations across a Mott gap or band transitions?

• Why would an excitation across a Mott gap show dispersion?

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La2CuO4 – Kim et al., 2002

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Light scattering processesIncoming photon wi Costs energy U

(charge transfer energy).

Electron hops, gains t.

Outgoing photon wf

For finite T, double occupancies lead to small band of low energy electrons.

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Metal-Insulator transition Falicov Kimball model d=∞

• Correlation-induced gap drives the single-particle DOS to zero at U=1.5

• Interacting DOS is independent of T in DMFT (Van Dongen, PRB, 1992)

• Examine Raman response through the (T=0) quantum phase transition.

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• Spectral weight shifts into charge transfer peak for increasing U.

• Low frequency spectral weight ~ t2/U.

Exact results: Falicov-Kimball

Charge transfer peaks.

Fixed Temperature

small band of

qps

Fixed U=2t

Charge transfer peaks.

Spectral weight

shifts into charge transfer peak for

increasing U or

decreasing T.

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Integrated spectral weight and inverse Raman slope

• The Raman response is sharply depleted

at low-T.

• The inverse Raman slope changes from nearly constant

uncorrelated metallic behavior to a rising

pseudogap or insulating behavior as

the correlations increase.

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Inelastic X-ray results U=4, n=1

• high energy peak – dispersionless charge transfer excitation ~ U.

• low energy peak is strongly temperature dependent.

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Peak positions and widthsLow energy peak High energy peak

Filled symbols – peak positions.

Open symbols – peak widths.

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Exact results for Hubbard model d=∞Nonresonant B1g Raman scattering

(n=1,U=2.1) • Note the charge transfer peak as well as the Fermi liquid peak at low energy. As T goes to zero, the Fermi peak sharpens and moves to lower energy.

• There is no low energy and low-T isosbestic point, rather a high frequency isosbestic point seems to develop.

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Nonresonant B1g Raman scattering (n=1,U=3.5)

• A MIT occurs as a function of T. Note the appearance of the low-T isosbestic point.

• The low energy Raman response has rich behavior, with a number of low energy peaks developing at low-T, but the low energy weight increases as T decreases.

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Nonresonant B1g Raman scattering (n=1,U=4.2)

• Universal behavior for the insulator---the low-energy spectral weight is depleted as T goes to zero and an isosbestic point appears.

• The temperature dependence here is over a wider range than for the FK model due to the T-dependence of the interacting DOS.

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X-ray results Hubbard Model

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Summary and Conclusions

• Shown some exact solutions for Raman scattering across a MIT.

• Insulating state, depletion of low energy spectral weight into charge transfer peak – universal behavior.

• Metallic state, development of low energy peak reflecting qp coherence.

• Elucidates dynamics near and through a quantum critical point.