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arXiv:cond-mat/0011232 · Physical Review Letters

Spin Orthogonality Catastrophe in Two-Dimensional Antiferromagnets and Superconductors

Subir Sachdev A, Matthias Troyer, Matthias Vojta C

Published Mar 19, 2001• 31 citations
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Abstract

We compute the spectral function of a spin S hole injected into a two-dimensional antiferromagnet or superconductor in the vicinity of a magnetic quantum critical point. We show that, near Van Hove singularities, the problem maps onto that of a static vacancy carrying excess spin S. The hole creation operator is characterized by a new boundary anomalous dimension and a vanishing quasiparticle residue at the critical point. We discuss possible relevance to photoemission spectra of cuprate superconductors near the antinodal points.

DOI
10.1103/physrevlett.86.2617

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The spectral function of a single hole moving in a two-dimensional antiferromagnetic background is calculated, using the self-consistent Born approximation for the self-energy. For small systems, good agreement is found with the results of exact diagonalization studies. In the thermodynamic limit, the quasiparticle residue, the effective mass, and the incoherent multiple spin-wave background are determined as functions of the ratio of Heisenberg exchange J and hopping amplitude t. The relevance of the single-hole results to the physical problem of a finite hole concentration is briefly discussed.

Physical review. B, Condensed matter•May 1, 1991•116 citations
We compute the spectral function of a spin S hole injected into a two-dimensional antiferromagnet or superconductor in the vicinity of a magnetic quantum critical point. We show that, near Van Hove singularities, the problem maps onto that of a static vacancy carrying excess spin S. The hole creation operator is characterized by a new boundary anomalous dimension and a vanishing quasiparticle residue at the critical point. We discuss possible relevance to photoemission spectra of cuprate superconductors near the antinodal points.
We calculate the angular resolved photoemission spectra (ARPES) of a spin-1/2 Heisenberg antiferromagnet as a function of magnetic fields using both the exact diagonalization method and self-consistent Born approximation. Below the saturation field BCB_CBC​, strong scattering between spin waves and a hole, created by photoemission of an electron, significantly narrows the quasiparticle band that is characterized by the lowering of the quasiparticle energy at (π,π)(π,π)(π,π) with increasing field. Accordingly, in ARPES the quasiparticle peak gets shaper near (π,π)(π,π)(π,π) and broader elsewhere. Furthermore, we observe that an anomalous extended van Hove region (EVHR) around (π,π)(π,π)(π,π) appears in a half saturation field, while EVHRs around (π,0)(π,0)(π,0) and (0,π)(0,π)(0,π) in zero field gradually disappear with increasing field.
The dynamical properties of hole motion in an antiferromagnetic background are determined in one-dimensional models in zero magnetic field, where spin isotropy holds, as well as in an external magnetic field. The latter case is also relevant, via particle-hole transformation, to the problem of hole propagation in one-dimensional ``superconductors.'' The singularities in the spectral function are investigated by means of bosonization techniques and perturbation theories. Results are then compared with Bethe ansatz solutions and Lanczos diagonalizations. The formalism also leads to interesting connections to the single impurity problem in Luttinger liquids. A rich structure is found in the spectral function whenever spin isotropy is broken, suggesting the presence of exotic momentum dependence in photoemission spectra of (quasi) one dimensional materials.
We study the hole and magnon spectral functions as a function of hole doping in the two-dimensional $t\ensuremath{-}J$ and $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}{t}^{\ensuremath{'}\ensuremath{'}}\ensuremath{-}J$ models working within the limits of spin-wave theory by linearizing the hole-spin-deviation interaction and by adapting the noncrossing approximation. We find that the staggered magnetization decreases rather rapidly with doping and it goes to zero at a few percent of hole concentration in both $t\ensuremath{-}J$ and $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}{t}^{\ensuremath{'}\ensuremath{'}}\ensuremath{-}J$ models. Furthermore, our results show that the residue of the quasiparticle peak at $\stackrel{P\vec}{G}=(\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2)$ decreases very rapidly with doping. We also find pockets centered at $\stackrel{P\vec}{G}$, (i) with an elliptical shape with large eccentricity along the antinodal direction in the case of the $t\ensuremath{-}J$ model and (ii) with an almost circular shape in the case of the $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}{t}^{\ensuremath{'}\ensuremath{'}}\ensuremath{-}J$ model. Last, we show that the spectral intensity distribution in the doped antiferromagnet has a waterfall-like pattern along the nodal direction of the Brillouin zone, a feature that is also seen in angle-resolved photoemission spectroscopy measurements.
We consider the two-dimensional $t\ensuremath{-}{t}^{\ensuremath{'}}$ Hubbard model with the Fermi level being close to Van Hove singularities. The phase diagram of the model is discussed. Far from the quantum phase transition (QPT) into ferro- (antiferro-) magnetic states, the self-energy at the singularity points has a nearly linear energy dependence, the density of states being proportional to ${\mathrm{ln}}^{3}|\ensuremath{\varepsilon}|.$ In the quantum-disordered and quantum-critical regimes near the QPT a scaling approach is used. At small energies the self-energy demonstrates a powerlike behavior with the exponent changing from unity to zero in a narrow energy interval. Provided that the system is close to ferromagnetic instability which occurs at ${t}^{\ensuremath{'}}/\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{-}1/2,$ the quasiparticle spectral weight has, instead of a narrow peak, a broad maximum and is nonmonotonous at small energies. The application of the results to cuprates is discussed.
Quasiparticle bands of the two-dimensional Hubbard model are calculated using the Roth two-pole approximation to the one-particle Green's function. Excellent agreement is obtained with recent Monte Carlo calculations, including an anomalous volume of the Fermi surface near half-filling, which can possibly be explained in terms of a breakdown of Fermi liquid theory. The calculated bands are very flat around the (\ensuremath{\pi},0) points of the Brillouin zone in agreement with photoemission measurements of cuprate superconductors. With doping there is a shift in spectral weight from the upper band to the lower band. The Roth method is extended to deal with superconductivity within a four-pole approximation allowing electron-hole mixing. It is shown that triplet p-wave pairing never occurs. A self-consistent solution with singlet ${\mathit{d}}_{{\mathit{x}}^{2}\mathrm{\ensuremath{-}}{\mathit{y}}^{2}}$-wave pairing is found and optimal doping occurs when the van Hove singularity, corresponding to the flat band part, lies at the Fermi level. Nearest-neighbor antiferromagnetic correlations play an important role in flattening the bands near the Fermi level and in favoring superconductivity. However, the mechanism for superconductivity is a local one, in contrast to spin-fluctuation exchange models. For reasonable values of the hopping parameter the transition temperature Tc{\mathit{T}}_{\mathit{c}}Tc​ is in the range 10--100 K. The optimum doping ${\mathrm{\ensuremath{\delta}}}_{\mathit{c}}$ lies between 0.14 and 0.25, depending on the ratio U/t. The gap equation has a BCS-like form and 2${\mathrm{\ensuremath{\Delta}}}_{\mathrm{max}}$/kTc{\mathit{kT}}_{\mathit{c}}kTc​\ensuremath{\simeq}4.
The spectral function of a single hole moving in a two-dimensional antiferromagnetic background is calculated, using the self-consistent Born approximation for the self-energy. For small systems, good agreement is found with the results of exact diagonalization studies. In the thermodynamic limit, the quasiparticle residue, the effective mass, and the incoherent multiple spin-wave background are determined as functions of the ratio of Heisenberg exchange J and hopping amplitude t. The relevance of the single-hole results to the physical problem of a finite hole concentration is briefly discussed.