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arXiv:0812.0593 · Physical Review B

Origin of the Mott gap

Robert G. Leigh, Philip Phillips

Published Jun 18, 2009• 14 citations
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Abstract

We show exactly that the only charged excitations that exist in the strong-coupling limit of the half-filled Hubbard model are gapped composite excitations generated by the dynamics of the charge $2e$ boson that appears upon explicit integration of the high-energy scale. At every momentum, such excitations have nonzero spectral weight at two distinct energy scales separated by the on-site repulsion $U$. The result is a gap in the spectrum for the composite excitations accompanied by a discontinuous vanishing of the density of states at the chemical potential when $U$ exceeds the bandwidth. Consequently, we resolve the long-standing problem of the cause of the charge gap in a half-filled band in the absence of symmetry breaking.

DOI
10.1103/physrevb.79.245120

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•Nov 8, 2011•0 citations
We show exactly that the only charged excitations that exist in the strong-coupling limit of the half-filled Hubbard model are gapped composite excitations generated by the dynamics of the charge 2e2e2e boson that appears upon explicit integration of the high-energy scale. At every momentum, such excitations have nonzero spectral weight at two distinct energy scales separated by the on-site repulsion UUU. The result is a gap in the spectrum for the composite excitations accompanied by a discontinuous vanishing of the density of states at the chemical potential when UUU exceeds the bandwidth. Consequently, we resolve the long-standing problem of the cause of the charge gap in a half-filled band in the absence of symmetry breaking.
We show here that many of the normal state properties of the cuprates can result from the new charge 2e2e2e bosonic field which we have recently [Phys. Rev. Lett. 99, 046404 (2007); Phys. Rev. B 77, 014512 (2007)] shown to exist in the exact low-energy theory of a doped Mott insulator. In particular, the (1) midinfrared band including the nonvanishing of the restricted fff-sum rule in the Mott insulator, (2) T2{T}^{2}T2 contribution to the thermal conductivity, (3) pseudogap, (4) bifurcation of the electron spectrum below the chemical potential, as recently seen in angle-resolved photoemission, (5) insulating behavior away from half-filling, (6) high- and low-energy kinks in the electron dispersion, and (7) TTT-linear resistivity all derive from the charge 2e2e2e bosonic field. We also calculate the inverse dielectric function and show that it possesses a sharp quasiparticle peak and a broad particle-hole continuum. The sharp peak is mediated by a new charge eee composite excitation formed from the binding of a charge 2e2e2e boson and a hole and represents a distinctly new prediction of this theory. It is this feature that is responsible for the dynamical part of the spectral weight transferred across the Mott gap. We propose that electron-energy loss spectroscopy at finite momentum and frequency can be used to probe the existence of such a sharp feature.
Though most fermionic Mott insulators order at low temperatures, ordering is ancillary to their insulating behavior. Our emphasis here is on disentangling ordering from the intrinsic strongly correlated physics of a doped half-filled band. To this end, we focus on the two-dimensional Hubbard model. Because the charge gap arises from on-site correlations, we have been refining the nonperturbative approach of Matsumoto and Mancini [Phys. Rev. B 55, 2095 (1997)] which incorporates local physics. Crucial to this method is a self-consistent two-site dynamical cluster expansion which builds in the nearest-neighbor energy scale J. At half-filling, we find that the spectral function possesses a gap of order U and is devoid of any coherent quasiparticle peaks although ordering or charge fractionalization are absent. At low temperatures, local antiferromagnetic correlations emerge. In the doped case, we find that the Fermi surface exceeds the Luttinger volume. The breakdown of Luttinger's theorem in the underdoped regime is traced both to the dynamically generated Mott gap as well to a nonvanishing of the imaginary part of the self-energy at the Fermi level. Spectral weight transfer across the Mott gap also emerges as a ubiquitous feature of a doped Mott insulator and suggests that high- and low-energy scales are inseparable. Additionally in the underdoped regime, we find that a pseudogap exists in the single-particle density of states as well as in the heat capacity. The pseudogap (which is set by the energy scale t2/U){t}^{2}/U)t2/U) is argued to be a ubiquitous feature of a lightly doped Mott state and simply represents the fact that hole transport involves double occupancy. In analogy with the Mott gap and antiferromagnetism, we propose that ordering may also accompany the formation of a pseudogap. We suggest a current pattern within a one-band model that preserves translational but breaks time-reversal symmetry along the canonical x and y axes but not along $x=\ifmmode\pm\else\textpm\fi{}y$ that is consistent with the experimental observations. Finally, we show that the Hall coefficient in a doped Mott insulator must change sign at a doping level x<1/3.x<1/3.x<1/3. The sign change is tied to a termination of strong correlation physics in the doped Mott state.
We expand on our earlier work [R. G. Leigh et al., Phys. Rev. Lett. 99, 46404 (2007)] in which we constructed the exact low energy theory of a doped Mott insulator by explicitly integrating (rather than projecting) out the degrees of freedom far away from the chemical potential. The exact low energy theory contains degrees of freedom that cannot be obtained from projective schemes. In particular, a charge $\ifmmode\pm\else\textpm\fi{}2e$ bosonic field that is not made out of elemental excitations emerges at low energies. Such a field accounts for dynamical spectral weight transfer across the Mott gap. At half-filling, we show that two such excitations emerge which play a crucial role in preserving the Luttinger surface along which the single-particle Green's function vanishes. In addition, the interactions with the bosonic fields defeat the artificial local SU(2) symmetry that is present in the Heisenberg model. We also apply this method to the Anderson-U impurity and show that in addition to the Kondo interaction, bosonic degrees of freedom appear as well. Finally, we show that as a result of the bosonic degree of freedom, the electron at low energies is in a linear superposition of two excitations---one arising from the standard projection into the low energy sector and the other from the binding of a hole and the boson.
With the aid of both a semianalytical and a numerically exact method, we investigate the charge dynamics in the vicinity of half-filling in the one- and two-dimensional $t\text{\ensuremath{-}}J$ model derived from a Fermi-Hubbard model in the limit of large interaction UUU and hence small exchange coupling JJJ. The spin degrees of freedom are taken to be disordered. So we consider the limit $0<J\ensuremath{\ll}T\ensuremath{\ll}W$, where WWW is the bandwidth. We focus on evaluating the local spectral density of a single hole excitation and the charge gap that separates the upper and the lower Hubbard band. We find indications that no band edges exist if the magnetic exchange is taken into account; instead of band edges, Gaussian tails seem to appear. A discussion of the underlying physics is provided.
We show that doped Mott insulators such as the copper-oxide superconductors are asymptotically slaved in that the quasiparticle weight Z near half-filling depends critically on the existence of the high-energy scale set by the upper Hubbard band. In particular, near half-filling, the following dichotomy arises: Z not equal to 0 when the high-energy scale is integrated out but Z=0 in the thermodynamic limit when it is retained. Slavery to the high-energy scale arises from quantum interference between electronic excitations across the Mott gap. Broad spectral features seen in photoemission in the normal state of the cuprates are argued to arise from high-energy slavery.
We study the charge dynamic structure factor of the one-dimensional Hubbard model with finite on-site repulsion U at half filling. Numerical results from the time-dependent density matrix renormalization group are analyzed by comparison with the exact spectrum of the model. The evolution of the line shape as a function of U is explained in terms of a relative transfer of spectral weight between the two-holon continuum that dominates in the limit U\to \infty and a subset of the two-holon-two-spinon continuum that reconstructs the electron-hole continuum in the limit U\to 0. Power-law singularities along boundary lines of the spectrum are described by effective impurity models that are explicitly invariant under spin and \eta-spin SU(2) rotations. The Mott-Hubbard metal-insulator transition is reflected in a discontinuous change of the exponents of edge singularities at U=0. The sharp feature observed in the spectrum for momenta near the zone boundary is attributed to a Van Hove singularity that persists as a consequence of integrability.