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arXiv:cond-mat/0301228 · Physical review. B, Condensed matter

Migdal’s theorem and the pseudogap

Philippe H. Monthoux

Published Aug 7, 2003• 18 citations
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Abstract

We study a model of quasiparticles on a two-dimensional square lattice coupled to Gaussian-distributed dynamical molecular fields. We consider two types of such fields, a vector molecular field that couples to the quasiparticle spin density and a scalar field coupled to the quasiparticle number density. The model describes quasiparticles coupled to spin or charge fluctuations, and is solved by a Monte Carlo sampling of the molecular-field distributions. When the molecular-field correlations are sufficiently weak, the corrections to the self-consistent Eliashberg theory do not bring about qualitative changes in the quasiparticle spectrum. But for a range of model parameters near the magnetic boundary, we find that Migdal's theorem does not apply and the quasiparticle spectrum is qualitatively different from its Eliashberg approximation. In the range of model parameters studied, we find that the transverse spin-fluctuation modes play a key role. While a pseudogap opens when quasiparticles are coupled to antiferromagnetic fluctuations, such a pseudogap is not observed in the corresponding charge-fluctuation case for the range of parameters studied, where vertex corrections are found to effectively reduce the strength of the interaction. This suggests that one has to be closer to the border of long-range order to observe pseudogap effects in the charge-fluctuation case than for a spin-fluctuation-induced interaction under otherwise similar conditions. An important feature of the magnetic pseudogap found in the present calculations is that it is strongly anisotropic. It vanishes along the diagonal of the Brillouin zone and is large near the zone boundary. In the case of ferromagnetic fluctuations, we also find a range of model parameters with qualitative changes in the quasiparticle spectral function not captured by the one-loop approximation, that is, the quasiparticle peak splits into two. We find that one needs to be closer to the magnetic boundary to observe the pseudogap effects in the nearly ferromagnetic case relative to the nearly antiferromagnetic one, under otherwise similar conditions. We provide intuitive arguments to explain the physical origin of the breakdown of Migdal's theorem.

DOI
10.1103/physrevb.68.064408

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We study a model of quasiparticles on a two-dimensional square lattice coupled to Gaussian-distributed dynamical molecular fields. We consider two types of such fields, a vector molecular field that couples to the quasiparticle spin density and a scalar field coupled to the quasiparticle number density. The model describes quasiparticles coupled to spin or charge fluctuations, and is solved by a Monte Carlo sampling of the molecular-field distributions. When the molecular-field correlations are sufficiently weak, the corrections to the self-consistent Eliashberg theory do not bring about qualitative changes in the quasiparticle spectrum. But for a range of model parameters near the magnetic boundary, we find that Migdal's theorem does not apply and the quasiparticle spectrum is qualitatively different from its Eliashberg approximation. In the range of model parameters studied, we find that the transverse spin-fluctuation modes play a key role. While a pseudogap opens when quasiparticles are coupled to antiferromagnetic fluctuations, such a pseudogap is not observed in the corresponding charge-fluctuation case for the range of parameters studied, where vertex corrections are found to effectively reduce the strength of the interaction. This suggests that one has to be closer to the border of long-range order to observe pseudogap effects in the charge-fluctuation case than for a spin-fluctuation-induced interaction under otherwise similar conditions. An important feature of the magnetic pseudogap found in the present calculations is that it is strongly anisotropic. It vanishes along the diagonal of the Brillouin zone and is large near the zone boundary. In the case of ferromagnetic fluctuations, we also find a range of model parameters with qualitative changes in the quasiparticle spectral function not captured by the one-loop approximation, that is, the quasiparticle peak splits into two. We find that one needs to be closer to the magnetic boundary to observe the pseudogap effects in the nearly ferromagnetic case relative to the nearly antiferromagnetic one, under otherwise similar conditions. We provide intuitive arguments to explain the physical origin of the breakdown of Migdal's theorem.
We study a model of quasiparticles on a two-dimensional square lattice coupled to Gaussian distributed dynamical molecular fields. We consider two types of such fields, a vector molecular field that couples to the quasiparticle spin density and a scalar field coupled to the quasiparticle number density. The model describes quasiparticles coupled to spin or charge fluctuations, and is solved by a Monte Carlo sampling of the molecular field distributions. The nonperturbative solution is compared to various approximations based on diagrammatic perturbation theory. When the molecular field correlations are sufficiently weak, the diagrammatic calculations capture the qualitative aspects of the quasiparticle spectrum. For a range of model parameters near the magnetic boundary, we find that the quasiparticle spectrum is qualitatively different from that of a Fermi liquid, in that it shows a double peak structure, and that the diagrammatic approximations we consider fail to reproduce, even qualitatively, the nonperturbative results of the Monte Carlo calculations. This suggests that the magnetic pseudogap induced by a coupling to antiferromagnetic spin fluctuations and the spin splitting of the quasiparticle peak induced by a coupling to ferromagnetic spin fluctuations lie beyond diagrammatic perturbation theory. While a pseudogap opens when quasiparticles are coupled to antiferromagnetic fluctuations, such a pseudogap is not observed in the corresponding charge-fluctuation case for the range of parameters studied, where vertex corrections are found to effectively reduce the strength of the interaction. This suggests that one has to be closer to the border of long-range order to observe pseudogap effects in the charge-fluctuation case than for a spin-fluctuation-induced interaction under otherwise similar conditions. The diagrammatic approximations that contain first-order vertex corrections show the enhancement of the spin-fluctuation-induced interaction and the suppression of the effective interaction in the charge-fluctuation case. However, for the range of model parameters considered here, the multiple spin or charge-fluctuation exchange processes not included in the diagrammatic approximations considered are found to be important, especially for quasiparticles coupled to charge fluctuations.
Pseudogap physics in strongly correlated systems is essentially scale dependent. We generalize the dynamical mean-field theory (DMFT) by introducing into the DMFT equations dependence on the correlation length of pseudogap fluctuations via an additional (momentum-dependent) self-energy Σk. This self-energy describes nonlocal dynamical correlations induced by short-ranged collective SDW-like antiferromagnetic spin (or CDW-like charge) fluctuations. At high enough temperatures these fluctuations can be viewed as a quenched Gaussian random field with finite correlation length. This generalized DMFT +Σk approach is used for the numerical solution of the weakly doped one-band Hubbard model with repulsive Coulomb interaction on a square lattice with nearest- and next-nearest-neighbor hopping. The effective single impurity problem is solved by the numerical renormalization group (NRG). Both types of strongly correlated metals, namely (i) the doped Mott insulator and (ii) the case of bandwidth W≲U (U is the value of local Coulomb interaction) are considered. Densities of states, spectral functions and ARPES spectra calculated within DMFT +Σk show a pseudogap formation near the Fermi level of the quasiparticle band. We also briefly discuss effects of random impurity scattering. Finally we demonstrate a qualitative picture of Fermi surface “destruction” due to pseudogap fluctuations and formation of “Fermi arcs” which agrees well with ARPES observations.
We generalize the dynamical-mean field (DMFT) approximation by including into the DMFT equations some length scale $\ensuremath{\xi}$ via a momentum dependent external self-energy ${\ensuremath{\Sigma}}_{\mathbf{k}}$. This external self-energy describes nonlocal dynamical correlations induced by the short-ranged collective spin density wave--like antiferromagnetic spin (or the charge density wave--like charge) fluctuations. At high enough temperatures these fluctuations can be viewed as a quenched Gaussian random field with a finite correlation length. This generalized $\mathrm{DMFT}+{\ensuremath{\Sigma}}_{\mathbf{k}}$ approach is used for the numerical solution of the weakly doped one-band Hubbard model with repulsive Coulomb interaction on a square lattice with the nearest and the next nearest neighbor hopping. The effective single impurity problem in this generalized $\mathrm{DMFT}+{\ensuremath{\Sigma}}_{\mathbf{k}}$ is solved by the numerical renormalization group. Both types of the strongly correlated metals, namely: (i) The doped Mott insulator and (ii) the case of the bandwidth $W\ensuremath{\lesssim}U$ (UUU---value of the local Coulomb interaction) are considered. The densities of states, the spectral functions, and the angle resolved photoemission spectra calculated within the $\mathrm{DMFT}+{\ensuremath{\Sigma}}_{\mathbf{k}}$ show a pseudogap formation near the Fermi level of the quasiparticle band.
Away from half filling and above its Neel temperature, the three-dimensional Hubbard model is a paramagnetic metal with antiferromagnetic spin fluctuations, where controlled momentum-resolved results have been lacking. Using connected-determinant diagrammatic Monte Carlo, we compute the momentum-resolved self-energy for n = 0.4-0.95, T/t = 0.05-0.33 and U/t = 4-8, and the full-zone spin susceptibility at twelve of these points, focusing on U/t = 4. At U/t = 4 and toward half filling the spin fluctuations become strong: for n = 0.7-0.9 at T/t = 0.14 they enhance the susceptibility peak up to sevenfold and raise the momentum-dependent part of the scattering rate to four to five times its second-order value. They are driven by an effective spin interaction screened to about 2U/3 and nearly uniform over the zone. Yet the peak height is set by the slightly smaller value of this interaction at the peak wave vector, and near magnetic order this small difference changes the height by a large factor. Two-particle self-consistent theory reproduces the peak because its sum rule fixes its single constant interaction near this value, but for n = 0.7-0.9 at T/t = 0.14 its self-energy overestimates the local scattering rate and underestimates the momentum-dependent part. Though strong, the fluctuations are short ranged, with a correlation length of at most about two lattice spacings, and for n = 0.7-0.9 and T/t >= 0.14 the scattering rate near the Fermi surface varies by at most about 1% at fixed band energy, without hot spots. At U/t = 4 dynamical mean-field theory (DMFT) reproduces the local scattering rate and quasiparticle weight within a few percent, and the nonlocal part it misses is approximately a function of the band energy. Strong spin fluctuations thus coexist with nearly isotropic quasiparticles, and the results provide a benchmark for extensions of DMFT and for spin-fluctuation theories.
The opening of a critical-fluctuation-induced pseudogap (or precursor pseudogap) in the one-particle spectral weight of the half-filled two-dimensional Hubbard model is discussed. This pseudogap, appearing in our Monte Carlo simulations, may be obtained from many-body techniques that use Green functions and vertex corrections that are at the same level of approximation. Self-consistent theories of the Eliashberg type (such as the fluctuation exchange approximation) use renormalized Green functions and bare vertices in a context where there is no Migdal theorem. They do not find the pseudogap, in quantitative and qualitative disagreement with simulations, suggesting these methods are inadequate for this problem. Differences between precursor pseudogaps and strong-coupling pseudogaps are also discussed.
A new approach to the single-band Hubbard model is described in the general context of many-body theories. It is based on enforcing conservation laws, the Pauli principle and a number of crucial sum-rules. More specifically, spin and charge susceptibilities are expressed, in a conserving approximation, as a function of two constant irreducible vertices whose values are found self-consistently. The Mermin-Wagner theorem in two dimensions is automatically satisfied. The effect of collective modes on single-particle properties is then obtained by a paramagnon-like formula that is consistent with the two-particle properties in the sense that the potential energy obtained from TrΣGTrΣGTrΣG is identical to that obtained using the fluctuation-dissipation theorem for susceptibilities. The vertex corrections are included through constant irreducible vertices. The theory is in quantitative agreement with Monte Carlo simulations for both single-particle and two-particle properties. In the two-dimensional renormalized classical regime, spin fluctuations lead to precursors of antiferromagnetic bands (shadow bands) and to the destruction of the Fermi-liquid quasiparticles in a wide temperature range above the zero-temperature phase transition. The analogous phenomenon of pairing pseudogap can occur in the attractive model in two dimensions when the pairing fluctuations become critical. Other many-body approaches are critically compared. It is argued that treating the spin fluctuations as if there was a Migdal's theorem can lead to wrong predictions, in particular with regard to the the single-particle pseudogap.