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Physical review. B, Condensed matter

Charged bosons in a doped Mott insulator: Electronic properties of domain-wall solitons and meron vortices

Mona Berciu, Sajeev John

Published Apr 15, 1998• 10 citations
Publisher

Abstract

We demonstrate from first principles that when a charge carrier is added to a $\mathrm{spin}\ensuremath{-}1/2$ antiferromagnetic Mott insulator in either a one- or two-dimensional lattice, the self-consistent, Hartree-Fock ground state consists of a magnetic soliton texture with a doubly degenerate electronic level at the center of the Mott-Hubbard charge gap. This model is appropriate to systems with weak interchain or interlayer magnetic couplings in which long-range antiferromagnetic order is observed in the absence of charge carriers (doping). These magnetic solitons mediate the destruction of the magnetic order as the charge carrier concentration is increased. In a one-dimensional lattice with nearest-neighbor hopping $t$, on-site Coulomb repulsion $U,$ and self-consistent, antiferromagnetic moment amplitude $S$, we find that a charged, fermionic, magnetic domain wall soliton with a weakly ferromagnetic core, centered between two sites, has lower Hartree-Fock energy than a corresponding charged quasiparticle in one of the Mott-Hubbard bands. However, for $US/t>2$, this soliton is unstable to the formation of a lower energy charged, bosonic domain wall soliton, centered on a single site. For $US/t<2$, both of the above solitons are charged bosons. The self-consistent structure of these solitons exhibits no rotation of the local magnetic moments, but only a local suppression of the local moment amplitude in the vicinity of the hole. In the absence of doping, charge neutral domain wall solitons exhibit spin rotation within their core region. The equilibrium core size $\ensuremath{\rho}$ is determined by the degree of magnetic anisotropy. The ferromagnetic core soliton exhibits a pair of nondegenerate near-midgap electronic states. The antiferromagnetic core soliton exhibits a pair of nondegenerate electronic states that are symmetric about the midgap energy and that merge into the continuum as the anisotropy effects are made small and the soliton core radius $\ensuremath{\rho}$ becomes very large. The two-dimensional antiferromagnetic Mott insulator exhibits analogous behavior to the one-dimensional model. This analogy is precise for a 2D antiferromagnet exhibiting spin flux. For the undoped Mott insulator, the ferromagnetic core meron vortex (``lotus flower'' configuration of local magnetic moments) exhibits a doubly degenerate electronic midgap state in the continuum model and is the analog of the 1D neutral domain wall soliton. We demonstrate that a hole added to the 2D system can form a charged bosonic collective excitation, in which the spin background around the hole forms a planar vortex with local antiferromagnetic correlations at infinity and vanishing local-moment amplitude at the core of the soliton.

DOI
10.1103/physrevb.57.9521

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We demonstrate from first principles that when a charge carrier is added to a spin\ensuremath−1/2\mathrm{spin}\ensuremath{-}1/2spin\ensuremath−1/2 antiferromagnetic Mott insulator in either a one- or two-dimensional lattice, the self-consistent, Hartree-Fock ground state consists of a magnetic soliton texture with a doubly degenerate electronic level at the center of the Mott-Hubbard charge gap. This model is appropriate to systems with weak interchain or interlayer magnetic couplings in which long-range antiferromagnetic order is observed in the absence of charge carriers (doping). These magnetic solitons mediate the destruction of the magnetic order as the charge carrier concentration is increased. In a one-dimensional lattice with nearest-neighbor hopping ttt, on-site Coulomb repulsion U,U,U, and self-consistent, antiferromagnetic moment amplitude SSS, we find that a charged, fermionic, magnetic domain wall soliton with a weakly ferromagnetic core, centered between two sites, has lower Hartree-Fock energy than a corresponding charged quasiparticle in one of the Mott-Hubbard bands. However, for US/t>2US/t>2US/t>2, this soliton is unstable to the formation of a lower energy charged, bosonic domain wall soliton, centered on a single site. For US/t<2US/t<2US/t<2, both of the above solitons are charged bosons. The self-consistent structure of these solitons exhibits no rotation of the local magnetic moments, but only a local suppression of the local moment amplitude in the vicinity of the hole. In the absence of doping, charge neutral domain wall solitons exhibit spin rotation within their core region. The equilibrium core size \ensuremathρ\ensuremath{\rho}\ensuremathρ is determined by the degree of magnetic anisotropy. The ferromagnetic core soliton exhibits a pair of nondegenerate near-midgap electronic states. The antiferromagnetic core soliton exhibits a pair of nondegenerate electronic states that are symmetric about the midgap energy and that merge into the continuum as the anisotropy effects are made small and the soliton core radius \ensuremathρ\ensuremath{\rho}\ensuremathρ becomes very large. The two-dimensional antiferromagnetic Mott insulator exhibits analogous behavior to the one-dimensional model. This analogy is precise for a 2D antiferromagnet exhibiting spin flux. For the undoped Mott insulator, the ferromagnetic core meron vortex (``lotus flower'' configuration of local magnetic moments) exhibits a doubly degenerate electronic midgap state in the continuum model and is the analog of the 1D neutral domain wall soliton. We demonstrate that a hole added to the 2D system can form a charged bosonic collective excitation, in which the spin background around the hole forms a planar vortex with local antiferromagnetic correlations at infinity and vanishing local-moment amplitude at the core of the soliton.
We demonstrate that the configuration interaction (CI) approximation recaptures essential features of the exact (Bethe-ansatz) solution to the one-dimensional (1D) Hubbard model. As such, it provides a valuable route for describing effects that go beyond mean-field theory for strongly correlated electron systems in higher dimensions. The CI method systematically describes fluctuation and quantum tunneling corrections to the Hartree-Fock approximation (HFA). HFA predicts that doping a half-filled Hubbard chain leads to the appearance of charged spin polarons or charged domain-wall solitons in the antiferromagnetic background. The CI method, on the other hand, describes the quantum dynamics of these charged magnetic solitons and quantum tunneling effects between various mean-field configurations. In this paper, we test the accuracy of the CI method against the exact solution of the one-dimensional Hubbard model. We find remarkable agreement between the energy of the mobile charged bosonic domain wall (as given by the CI method) and the exact energy of the doping hole (as given by the Bethe ansatz) for the entire U/tU/tU/t range. The CI method also leads to a clear demonstration of the spin-charge separation in one dimension. Addition of one doping hole to the half-filled antiferromagnetic chain results in the appearance of two different carriers: a charged bosonic domain wall (which carries the charge but no spin) and a neutral spin-1/2 domain wall (which carries the spin but no charge).
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.
We investigate a microscopic model for strongly correlated electrons with both on-site and nearest-neighbor Coulomb repulsion on a two-dimensional (2D) square lattice. This exhibits a state in which electrons undergo a ``somersault'' in their internal spin space (spin flux) as they traverse a closed loop in external coordinate space. When this spin-12\frac{1}{2}21​ antiferromagnetic (AFM) insulator is doped, the ground state is a liquid of charged, bosonic meron vortices, which for topological reasons are created in vortex-antivortex pairs. The magnetic exchange energy of the distorted AFM background leads to a logarithmic vortex-antivortex attraction which overcomes the direct Coulomb repulsion between holes localized on the vortex cores. This leads to the appearance of preformed charged pairs. We use the configuration interaction (CI) method to study the quantum translational and rotational motion of various charged magnetic solitons and soliton pairs. The CI method systematically describes fluctuation and quantum tunneling corrections to the Hartree-Fock approximation. We find that the lowest-energy charged meron-antimeron pairs exhibit d-wave rotational symmetry, consistent with the symmetry of the cuprate superconducting order parameter. For a single hole in the 2D AFM plane, we find a precursor to spin-charge separation in which a conventional charged spin polaron dissociates into a singly charged meron-antimeron bound pair. This model provides a unified microscopic basis for (i) non-Fermi-liquid transport properties, (ii) d-wave preformed charged carrier pairs, (iii) midinfrared optical absorption, (iv) destruction of AFM long-range order with doping and other magnetic properties, and (v) certain aspects of angle-resolved photoemission spectroscopy.
We determine the spin and charge orders in the ground state of the doped two-dimensional (2D) Hubbard model in its simplest form, namely with only nearest-neighbor hopping and on-site repulsion. At half-filling, the ground state is known to be an antiferromagnetic Mott insulator. Doping Mott insulators is believed to be relevant to the superconductivity observed in cuprates. A variety of candidates have been proposed for the ground state of the doped 2D Hubbard model. A recent work employing a combination of several state-of-the-art numerical many-body methods established the stripe order as the ground state near 1/81/81/8 doping at strong interactions. In this paper, we apply one of these methods, the cutting-edge constrained-path auxiliary field quantum Monte Carlo (AFQMC) method with self-consistently optimized gauge constraints, to systematically study the model as a function of doping and interaction strength. With careful finite size scaling based on large-scale computations, we map out the ground state phase diagram in terms of its spin and charge order. We find that modulated antiferromagnetic order persists from near half-filling to about 1/51/51/5 doping. At lower interaction strengths or larger doping, these ordered states are best described as spin-density waves, with essentially delocalized holes and modest oscillations in charge correlations. When the charge correlations are stronger (large interaction or small doping), they are best described as stripe states, with the holes more localized near the node in the antiferromagnetic spin order. In both cases, we find that the wavelength in the charge correlations is consistent with so-called filled stripes.
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 describe a new microscopic approach for analyzing interacting electron systems with local moments or, in principle, any local order parameter. We specialize attention to the doped Mott insulator phase of the Hubbard model, where standard weak-coupling perturbation methods fail. A rotationally invariant Stratonovich-Hubbard field is introduced to decouple the static spin components of the interaction at each site. Charge degrees of freedom are then treated by ``slave'' Hartree-Fock in the presence of a spatially varying random spin field. The effective action reduces to a classical Heisenberg model at half filling, insuring that the system has (i) finite-range order at T>0 with an exponentially diverging correlation length and (ii) a one-electron Mott-Hubbard gap in the presence of disorder. Away from half-filling properties are determined by strongly non-Gaussian fluctuations in the amplitude and orientation of the local spin fields. The saddle point equations of the theory at zero temperature reduce to inhomogeneous Hartree-Fock, so that disordering of domain walls at finite temperature is in principle included. We present preliminary small system results for the intermediate and large interaction regimes obtained by Monte Carlo simulation of random spin field configurations.