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arXiv:cond-mat/0607693 · Physical Review B

Theory of core-level photoemission and the x-ray edge singularity across the Mott transition

Pablo S. Cornaglia, Antoine Georges

Published Mar 12, 2007• 29 citations
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Abstract

The zero-temperature core-level photoemission spectrum of a Hubbard system is studied across the metal to Mott insulator transition using dynamical mean-field theory and Wilson's numerical renormalization group [Rev. Mod. Phys. 47, 773 (1975)]. An asymmetric power-law divergence is obtained in the metallic phase with an exponent $\ensuremath{\alpha}(U,Q)\ensuremath{-}1$, which depends on the strength of both the Hubbard interaction $U$ and the core-hole potential $Q$. For $Q\ensuremath{\lesssim}{U}_{c}∕2$, $\ensuremath{\alpha}$ decreases with increasing $U$ and vanishes at the transition $(U\ensuremath{\rightarrow}{U}_{c})$ leading to a symmetric peak in the insulating phase. For $Q\ensuremath{\gtrsim}{U}_{c}∕2$, $\ensuremath{\alpha}$ remains finite, close to the transition, but the integrated intensity of the power law vanishes and there is no associated peak in the insulator. The weight and position of the remaining peaks in the spectra can be understood within a molecular orbital approach.

DOI
10.1103/physrevb.75.115112

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The zero-temperature core-level photoemission spectrum of a Hubbard system is studied across the metal to Mott insulator transition using dynamical mean-field theory and Wilson's numerical renormalization group [Rev. Mod. Phys. 47, 773 (1975)]. An asymmetric power-law divergence is obtained in the metallic phase with an exponent \ensuremathα(U,Q)\ensuremath−1\ensuremath{\alpha}(U,Q)\ensuremath{-}1\ensuremathα(U,Q)\ensuremath−1, which depends on the strength of both the Hubbard interaction UUU and the core-hole potential QQQ. For Q\ensuremath≲Uc∕2Q\ensuremath{\lesssim}{U}_{c}∕2Q\ensuremath≲Uc​∕2, \ensuremathα\ensuremath{\alpha}\ensuremathα decreases with increasing UUU and vanishes at the transition (U\ensuremath→Uc)(U\ensuremath{\rightarrow}{U}_{c})(U\ensuremath→Uc​) leading to a symmetric peak in the insulating phase. For Q\ensuremath≳Uc∕2Q\ensuremath{\gtrsim}{U}_{c}∕2Q\ensuremath≳Uc​∕2, \ensuremathα\ensuremath{\alpha}\ensuremathα remains finite, close to the transition, but the integrated intensity of the power law vanishes and there is no associated peak in the insulator. The weight and position of the remaining peaks in the spectra can be understood within a molecular orbital approach.
Wilson's numerical renormalization group method for the calculation of dynamic properties of impurity models is generalized to investigate the effective impurity model of the dynamical mean-field theory at finite temperatures. We calculate the spectral function and self-energy for the Hubbard model on a Bethe lattice with infinite coordination number directly on the real-frequency axis and investigate the phase diagram for the Mott-Hubbard metal-insulator transition. While for $T<{T}_{\mathrm{c}}\ensuremath{\approx}0.02W$ (W:(W:(W: bandwidth) we find hysteresis with first-order transitions both at Uc1{U}_{\mathrm{c}1}Uc1​ (defining the insulator to metal transition) and at Uc2{U}_{\mathrm{c}2}Uc2​ (defining the metal to insulator transition), at T>TcT>{T}_{\mathrm{c}}T>Tc​ there is a smooth crossover from metalliclike to insulatinglike solutions.
The fff-electron spectral function of the Falicov-Kimball model is calculated within the dynamical mean-field theory using the numerical renormalization group method as the impurity solver. Both the Bethe lattice and the hypercubic lattice are considered at half filling. For small UUU we obtain a single-peaked fff-electron spectral function, which---for zero temperature---exhibits an algebraic (x-ray) singularity $({\ensuremath{\mid}\ensuremath{\omega}\ensuremath{\mid}}^{\ensuremath{-}\ensuremath{\alpha}})$ for $\ensuremath{\omega}\ensuremath{\rightarrow}0$. The characteristic exponent $\ensuremath{\alpha}$ depends on the Coulomb (Hubbard) correlation UUU. This x-ray singularity cannot be observed when using alternative (Keldysh-based) many-body approaches. With increasing $U,\ensuremath{\alpha}$ decreases and vanishes for sufficiently large UUU when the fff-electron spectral function develops a gap and a two-peak structure (metal-insulator transition).
Recent studies of electrical transport, both theoretical and experimental, near the bandwidth-tuned Mott metal-insulator transition have uncovered apparent quantum critical scaling of the electrical resistivity at elevated temperatures, despite the fact that the actual low-temperature phase transition is of first order. This raises the question whether there is a hidden Mott quantum critical point. Here we argue that the dynamical mean-field theory of the Hubbard model admits, in the low-temperature limit, asymptotically scale-invariant (i.e., power-law) solutions, corresponding to the metastable insulator at the boundary of the metal-insulator coexistence region; these solutions can be linked to the physics of the pseudogap Anderson model. While our state-of-the-art numerical renormalization-group calculations reveal that this asymptotic regime is restricted to very small energies and temperatures and hence is difficult to access numerically, we uncover the existence of a wide crossover regime where the single-particle spectrum displays a different power law. We show that it is this power-law regime, corresponding to approximate local quantum criticality, which is continuously connected to and responsible for the apparent quantum critical scaling above the classical critical end point. We connect our findings to experiments on tunable Mott materials.
The paramagnetic phase of the one-band Hubbard model is studied at zero-temperature, within the framework of dynamical mean-field theory, and for general particle-hole asymmetry where a doping-induced Mott transition occurs. Our primary focus is the Mott insulator (MI) phase, and our main aim to establish what can be shown exactly about it. To handle the locally doubly-degenerate MI requires two distinct self-energies, which reflect the broken symmetry nature of the phase and together determine the standard single self-energy. Exact results are obtained for the local charge, local magnetic moment and associated spin susceptibilities, the interaction-renormalised levels, and the low-energy behaviour of the self-energy in the MI phase. The metallic phase is also considered briefly, and shown to acquire an emergent particle-hole symmetry as the Mott transition is approached. Throughout the metal, Luttinger's theorem is reflected in the vanishing of the Luttinger integral; for the generic MI by contrast this is shown to be non-vanishing, but again to have a universal magnitude. Numerical results are also obtained using NRG, for the metal/MI phase boundary, the scaling behaviour of the charge as the Mott transition is aproached from the metal, and associated universal scaling of single-particle dynamics as the low-energy Kondo scale vanishes.
Correlated electrons in a binary alloy ${A}_{x}{B}_{1\ensuremath{-}x}$ are investigated within the Hubbard model and dynamical mean-field theory (DMFT). The random energies ${\ensuremath{\epsilon}}_{i}$ have a bimodal probability distribution and an energy separation $\ensuremath{\Delta}.$ We solve the DMFT equations by the numerical renormalization-group method at zero temperature and calculate the spectral density as a function of disorder strength $\ensuremath{\Delta}$ and interaction U at different fillings. For filling factors $\ensuremath{\nu}=x$ or 1+x1+x1+x the lower or upper alloy subband is half filled and the system becomes a Mott insulator at strong interactions, with a correlation gap at the Fermi level. At the metal-insulator transition hysteresis is observed. We also analyze the effective theory in the $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Delta}}\ensuremath{\infty}$ limit and find good agreement between analytical and numerical results for the critical interaction Uc{U}_{c}Uc​ at which the metal-insulator transition occurs.