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arXiv:cond-mat/0501331 · Physical Review B

Electronic self-energy and triplet pairing fluctuations in the vicinity of a ferromagnetic instability in two-dimensional systems: Quasistatic approach

A. A. Katanin

Published Jul 8, 2005• 15 citations
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Abstract

The self-energy, spectral functions, and susceptibilities of two-dimensional systems with strong ferromagnetic fluctuations are considered within the quasistatic approach. The self-energy at low temperatures $T$ has a non-Fermi-liquid form in the energy window $\ensuremath{\mid}\ensuremath{\omega}\ensuremath{\mid}\ensuremath{\lesssim}{\ensuremath{\Delta}}_{0}$ near the Fermi level, where ${\ensuremath{\Delta}}_{0}$ is the ground-state spin splitting for magnetically ordered ground state, and ${\ensuremath{\Delta}}_{0}\ensuremath{\propto}{T}^{1∕2}\phantom{\rule{0.2em}{0ex}}{\mathrm{ln}}^{1∕2}({v}_{F}∕T)$ in the quantum critical regime (${v}_{F}$ is the Fermi velocity). Spectral functions have a two-peak structure at finite $T$ above the magnetically ordered ground state, which implies quasisplitting of the Fermi surface in the paramagnetic phase in the presence of strong ferromagnetic fluctuations. The triplet pairing amplitude in the quasistatic approximation increases with increasing correlation length; at low temperatures $T⪡{\ensuremath{\Delta}}_{0}$ the vertex corrections become important and the Eliashberg approach is not justified. The results for the spectral properties and susceptibilities in the quantum critical regime near charge (spin) instabilities with large enough correlation length $\ensuremath{\xi}⪢{(T∕{v}_{F})}^{\ensuremath{-}1∕3}$ are obtained.

DOI
10.1103/physrevb.72.035111

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We consider the electronic and magnetic properties of the Hubbard model on a square lattice with the Fermi level near van Hove singularity and the ratio of the next-nearest-neighbor and nearest-neighbor hoppings t′/t=−0.45t'/t=-0.45t′/t=−0.45, which favours the ferromagnetic instability. We find, that a self-consistent consideration of the ferromagnetic fluctuations within the D-TRILEX approach results in the splitting of the electronic spectral function at low temperatures. This splitting exhibits only a weak momentum dependence, and only one of the split bands crosses the Fermi level. As a result, the Fermi surface itself remains unsplit, but its area increases, reflecting the presence of non-Fermi-liquid electronic excitations. We show that both the self-consistent account of the non-local contributions to the electronic self-energy and the proper treatment of electron interaction vertices in D-TRILEX are important to obtain this behaviour.
Based on recent experimental evidence for a spin-polarized ground state in the insulating phase of the two-dimensional electron system, we propose that ferromagnetic spin fluctuations lead to an attractive interaction in the triplet channel and cause p-wave pairing in the conducting phase. We use the Landau-Fermi liquid phenomenology to explain how the enhanced spin susceptibility near the critical density yields an attractive potential, in a similar mechanism to superfluidity in 3He.{}^{3}\mathrm{He}.3He. As the density is decreased, the p-wave order parameter undergoes a transition from a unitary to a nonunitary state, in which it coexists with ferromagnetism for a range of densities. As the density is further reduced, the pairing amplitude vanishes and the system is described by a ferromagnetic insulator. Thus, we find two quantum critical points as a function of density associated with the polarization of the paired state and ferromagnetism. We explain the magnetotransport measurements in parallel and perpendicular magnetic fields and propose a shot-noise experiment to measure the pair charge.
We present the expression for the quasiparticle vertex function ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ (proportional to the Landau interaction function) in a 2D Fermi liquid (FL) near an instability towards antiferromagnetism. This function is relevant in many ways in the context of metallic quantum criticality. Previous studies have found that near a quantum critical point, the system enters into a regime in which the fermionic self-energy is large near hot spots on the Fermi surface [points on the Fermi surface connected by the antiferromagnetic ordering vector ${q}_{\ensuremath{\pi}}=(\ensuremath{\pi},\ensuremath{\pi})$] and has much stronger dependence on frequency than on momentum. We show that in this regime, which we termed a critical FL, the conventional random-phase-approximation- (RPA) type approach breaks down, and to properly calculate the vertex function one has to sum up an infinite series of terms which were explicitly excluded in the conventional treatment. Besides, we show that, to properly describe the spin component of ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ even in an ordinary FL, one has to add Aslamazov-Larkin (AL) terms to the RPA vertex. We show that the total ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ is larger in a critical FL than in an ordinary FL, roughly by an extra power of magnetic correlation length $\ensuremath{\xi}$, which diverges at the quantum critical point. However, the enhancement of ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ is highly nonuniform: It holds only when, for one of the two momentum variables, the distance from a hot spot along the Fermi surface is much larger than for the other one. This fact renders our case different from quantum criticality at small momentum, where the enhancement of ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ was found to be homogeneous. We show that the charge and spin components of the total vertex function satisfy the universal relations following from the Ward identities related to the conservation of the particle number and the total spin. We show that in a critical FL, the Ward identity involves ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ taken between particles on the FS. We find that the charge and spin components of ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ are identical to leading order in the magnetic correlation length. We use our results for ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ and for the quasiparticle residue to derive the Landau parameters Fcl=0=Fsl=0{F}_{c}^{l=0}={F}_{s}^{l=0}Fcl=0​=Fsl=0​, the density of states, and the uniform (q=0q=0q=0) charge and spin susceptibilities ${\ensuremath{\chi}}_{c}^{l=0}={\ensuremath{\chi}}_{s}^{l=0}$. We show that the density of states NF{N}_{F}NF​ diverges as $log\ensuremath{\xi}$; however, Fc,sl=0{F}_{c,s}^{l=0}Fc,sl=0​ also diverge as $log\ensuremath{\xi}$, such that the total ${\ensuremath{\chi}}_{c,s}^{(l=0)}\ensuremath{\propto}{N}_{F}/(1+{F}_{c}^{l=0})$ remain finite at $\ensuremath{\xi}=\ensuremath{\infty}$. We show that at weak coupling these susceptibilities are parametrically smaller than for free fermions.
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We have developed a semi-analytical framework formulated in the canonical fermion representation to investigate strongly correlated electron systems. We consider the U=∞\infty∞ Hubbard model and used the equation of motion method to calculate the fermion self-energy which has two parts: single and two-boson exchange processes. The emergent bosons here are self-generated local charge and spin-density fluctuations which become strongly time-dependent due to extreme correlations. The computed boson spectral density is a diffusive damped mode with a long tail. The electron self-energy at d=∞d=\inftyd=∞ is computed self-consistently. The corresponding fermionic spectral density displays a pronounced coherence peak at ω=0ω=0ω=0, while its frequency derivative develops a two-peak structure at finite ωωω. The resistivity shows a linear temperature dependence over a broad range, crossing over to coherent Fermi-liquid behavior at extremely low temperatures.
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.