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arXiv:0707.1554 · Physical Review B

Exact integration of the high energy scale in doped Mott insulators

Ting-Pong Choy, Robert G. Leigh, Philip Phillips, Philip D. Powell

Published Jan 17, 2008• 21 citations
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Abstract

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.

DOI
10.1103/physrevb.77.014512

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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±\else\textpm\fi2e\ifmmode\pm\else\textpm\fi{}2e\ifmmode±\else\textpm\fi2e 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.
We construct the low-energy theory of a doped Mott insulator, such as the high-temperature superconductors, by explicitly integrating over the degrees of freedom far away from the chemical potential. For either hole or electron doping, a charge 2e bosonic field emerges at low energy. The charge 2e boson mediates dynamical spectral weight transfer across the Mott gap and creates a new charge e excitation by binding a hole. The result is a bifurcation of the electron dispersion below the chemical potential as observed recently in angle-resolved photoemission on Pb-doped Bi2Sr2CaCu2O8+delta (Pb2212).
We show that doped Mott insulators exhibit a collective degree of freedom, not made out of the elemental excitations, because the number of single-particle addition states at low energy per electron per spin is greater than one. The presence of such a collective degree of freedom which is not a consequence of proximity to a phase transition is a consequence of dynamical spectral weight transfer from high to low energies. This physics is captured by the charge 2e2e2e boson that emerges by explicitly integrating out the high-energy scale in the Hubbard model. The charge 2e2e2e boson binds to a hole, thereby mediating new charge eee states at low energy. It is the presence of such charge eee states which have no counterpart in the non-interacting system that provides the general mechanism for the breakdown of Fermi liquid theory in doped Mott insulators. The relationship between the charge 2e2e2e boson formulation and the standard perturbative treatment is explained.
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 introduce a slave-fermion formulation in which to study the charge dynamics of the half-filled Hubbard model on the square lattice. In this description, the charge degrees of freedom are represented by fermionic holons and doublons and the Mott-insulating characteristics of the ground state are the consequence of holon–doublon bound-state formation. The bosonic spin degrees of freedom are described by the antiferromagnetic Heisenberg model, yielding long-ranged (Néel) magnetic order at zero temperature. Within this framework and in the self-consistent Born approximation, we perform systematic calculations of the average double occupancy, the electronic density of states, the spectral function and the optical conductivity. Qualitatively, our method reproduces the lower and upper Hubbard bands, the spectral-weight transfer into a coherent quasiparticle band at their lower edges and the renormalisation of the Mott gap, which is associated with holon–doublon binding, due to the interactions of both quasiparticle species with the magnons. The zeros of the Green function at the chemical potential give the Luttinger volume, the poles of the self-energy reflect the underlying quasiparticle dispersion with a spin-renormalised hopping parameter and the optical gap is directly related to the Mott gap. Quantitatively, the square-lattice Hubbard model is one of the best-characterised problems in correlated condensed matter and many numerical calculations, all with different strengths and weaknesses, exist with which to benchmark our approach. From the semi-quantitative accuracy of our results for all but the weakest interaction strengths, we conclude that a self-consistent treatment of the spin-fluctuation effects on the charge degrees of freedom captures all the essential physics of the antiferromagnetic Mott–Hubbard insulator. We remark in addition that an analytical approximation with these properties serves a vital function in developing a full understanding of the fundamental physics of the Mott state, both in the antiferromagnetic insulator and at finite temperatures and dopings.
With the hierarchical Green’s function approach, we study a doped Mott insulator described with the Hubbard model by analytically solving the equations of motion of an one-particle Green’s function and related multiple-point correlation functions, and find that the separation of the spin and charge degrees of freedom of the electrons is an intrinsic character of the doped Mott insulator. For enough of large on-site repulsive Coulomb interaction, we show that the spectral weight of the one-particle Green’s function is proportional to the hole doping concentration that is mainly produced by the charge fluctuation of electrons, while the excitation spectrum of the electrons is composed of two parts: One is contributed by the spin fluctuation of the electrons, which is proportional to the hole doping concentration, and another one is coming from the coupling between the charge and spin fluctuations of the electrons that takes the maximum at undoping. All of these low energy/temperature physical properties originate from the strong on-site Coulomb interaction. The present results are consistent with the spectroscopy observations of the cuprate superconductors and the numerical calculations in normal state above the pseudogap regime.