We present the full analysis of the normal state properties of the spin-fermion model near the antiferromagnetic instability in two dimensions. The model describes low-energy fermions interacting with their own collective spin fluctuations, which soften at the antiferromagnetic transition. We argue that in 2D, the system has two typical energies—an effective spin-fermion interaction g¯ and an energy ωsf below which the system behaves as a Fermi liquid. The ratio of the two determines the dimensionless coupling constant for spin-fermion interaction λ2 ∝ g¯/ωsf. We show that λ scales with the spin correlation length and diverges at criticality. This divergence implies that the conventional perturbative expansion breaks down. We develop a novel approach to the problem—the expansion in either the inverse number of hot spots in the Brillouin zone, or the inverse number of fermionic flavours—which allows us to explicitly account for all terms which diverge as powers of λ, and treat the remaining, O(log λ) terms in the RG formalism. We apply this technique to study the properties of the spin-fermion model in various frequency and temperature regimes. We present the results for the fermionic spectral function, spin susceptibility, optical conductivity and other observables. We compare our results in detail with the normal state data for the cuprates, and argue that the spin-fermion model is capable of explaining the anomalous normal state properties of the high T c materials. We also show that the conventional {4 theory of the quantum-critical behaviour is inapplicable in 2D due to the singularity of the {4 vertex.
We present the full analysis of the normal state properties of the spin-fermion model near the antiferromagnetic instability in two dimensions. The model describes low-energy fermions interacting with their own collective spin fluctuations, which soften at the antiferromagnetic transition. We argue that in 2D, the system has two typical energies—an effective spin-fermion interaction g¯ and an energy ωsf below which the system behaves as a Fermi liquid. The ratio of the two determines the dimensionless coupling constant for spin-fermion interaction λ2 ∝ g¯/ωsf. We show that λ scales with the spin correlation length and diverges at criticality. This divergence implies that the conventional perturbative expansion breaks down. We develop a novel approach to the problem—the expansion in either the inverse number of hot spots in the Brillouin zone, or the inverse number of fermionic flavours—which allows us to explicitly account for all terms which diverge as powers of λ, and treat the remaining, O(log λ) terms in the RG formalism. We apply this technique to study the properties of the spin-fermion model in various frequency and temperature regimes. We present the results for the fermionic spectral function, spin susceptibility, optical conductivity and other observables. We compare our results in detail with the normal state data for the cuprates, and argue that the spin-fermion model is capable of explaining the anomalous normal state properties of the high T c materials. We also show that the conventional {4 theory of the quantum-critical behaviour is inapplicable in 2D due to the singularity of the {4 vertex.
We consider spin and electronic properties of itinerant electron systems, described by the spin-fermion model, near the antiferromagnetic critical point. We expand in the inverse number of hot spots in the Brillouin zone, N, and present the results beyond the previously studied $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\infty}$ limit. We found two new effects: (i) Fermi surface becomes nested at hot spots, and (ii) vertex corrections give rise to anomalous spin dynamics and change the dynamical critical exponent from z=2 to z>2. To first order in 1/N we found $z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2N/(N\ensuremath{-}2)$ which for a physical N=8 yields $z\ensuremath{\approx}2.67$.
Many correlated materials display a quantum-critical point between a paramagnetic and a spin-density wave (SDW) state. The SDW wave vector connects points, so-called hot spots, on opposite sides of the Fermi surface. The Fermi velocities at these pairs of points are in general not parallel. Here, we consider the case where pairs of hot spots coalesce, and the wave vector $(\ensuremath{\pi},\ensuremath{\pi})$ of the SDW connects hot spots with parallel Fermi velocities. Using the specific example of electron-doped cuprates, we first show that Kanamori screening and generic features of the Lindhard function make this case experimentally relevant. The temperature dependence of the correlation length, the spin susceptibility, and the self-energy at the hot spots are found using the two-particle self-consistent theory and specific numerical examples worked out for band and interaction parameters characteristic of the electron-doped cuprates. While the curvature of the Fermi surface at the hot spots leads to deviations from perfect nesting, the pseudonesting conditions lead to drastic modifications of the temperature dependence of these physical observables: Neglecting logarithmic corrections, the correlation length $\ensuremath{\xi}$ scales like 1/T, namely, z=1 instead of the naive z=2, the $(\ensuremath{\pi},\ensuremath{\pi})$ static spin susceptibility $\ensuremath{\chi}$ like 1/T, and the imaginary part of the self-energy at the hot spots like T3/2. The correction ${T}_{1}^{\ensuremath{-}1}\ensuremath{\sim}{T}^{3/2}$ to the Korringa NMR relaxation rate is subdominant. We also consider this problem at zero temperature, or for frequencies larger than temperature, using a field-theoretical model of gapless collective bosonic modes (SDW fluctuations) interacting with fermions. The imaginary part of the retarded fermionic self-energy close to the hot spots scales as $\ensuremath{-}{\ensuremath{\omega}}^{3/2}\mathrm{ln}\ensuremath{\omega}$. This is less singular than earlier predictions of the form $\ensuremath{-}\ensuremath{\omega}\mathrm{ln}\ensuremath{\omega}$. The difference arises from the effects of umklapp terms that were not included in previous studies.
Metallic quantum criticality is among the central themes in the understanding of correlated electronic systems, and converging results between analytical and numerical approaches are still under review. In this work, we develop a state-of-the-art large-scale quantum Monte Carlo simulation technique and systematically investigate the itinerant quantum critical point on a 2D square lattice with antiferromagnetic spin fluctuations at wavevector [Formula: see text]-a problem that resembles the Fermi surface setup and low-energy antiferromagnetic fluctuations in high-Tc cuprates and other critical metals, which might be relevant to their non-Fermi-liquid behaviors. System sizes of [Formula: see text] ([Formula: see text]) are comfortably accessed, and the quantum critical scaling behaviors are revealed with unprecedented high precision. We found that the antiferromagnetic spin fluctuations introduce effective interactions among fermions and the fermions in return render the bare bosonic critical point into a different universality, different from both the bare Ising universality class and the Hertz-Mills-Moriya RPA prediction. At the quantum critical point, a finite anomalous dimension [Formula: see text] is observed in the bosonic propagator, and fermions at hotspots evolve into a non-Fermi liquid. In the antiferromagnetically ordered metallic phase, fermion pockets are observed as the energy gap opens up at the hotspots. These results bridge the recent theoretical and numerical developments in metallic quantum criticality and can serve as the stepping stone toward final understanding of the 2D correlated fermions interacting with gapless critical excitations.
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, the density of states, and the uniform (q=0) charge and spin susceptibilities ${\ensuremath{\chi}}_{c}^{l=0}={\ensuremath{\chi}}_{s}^{l=0}$. We show that the density of states NF diverges as $log\ensuremath{\xi}$; however, 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.
Kinetic properties of a two-dimensional model of fermions interacting with antiferromagnetic spin excitations near the quantum critical point are considered. The temperature or doping are assumed to be sufficiently high, such that the pseudogap does not appear. In contrast to standard spin-fermion models, it is assumed that there are intrinsic inhomogeneities in the system suppressing space correlations of the antiferromagnetic excitations. It is argued that the inhomogeneities in the spin excitations in the ``strange metal'' phase can be a consequence of the existence of ``$\ensuremath{\pi}$-shifted'' domain walls in the doped antiferromagnetic phase. Averaging over the inhomogeneities and calculating physical quantities such as resistivity and some others one can explain unusual properties of cuprates unified under the name ``marginal Fermi liquid.'' The dependence of the slope of the linear temperature dependence of the resistivity on doping is compared with experimental data.
We consider two-dimensional Fermi liquids in the vicinity of a quantum transition to a phase with commensurate, antiferromagnetic long-range order. Depending upon the Fermi-surface topology, mean-field spin-density-wave theory predicts two different types of such transitions, with mean-field dynamic critical exponents z=1 (when the Fermi surface does not cross the magnetic zone boundary, type A) and z=2 (when the Fermi surface crosses the magnetic zone boundary, type B). The type-A system only displays z=1 behavior at all energies and its scaling properties are similar (though not identical) to those of an insulating Heisenberg antiferromagnet. Under suitable conditions precisely stated in this paper, the type-B system displays a crossover from relaxational behavior at low energies to type-A behavior at high energies. A scaling hypothesis is proposed to describe this crossover: we postulate a universal scaling function which determines the entire, temperature-, wave-vector-, and frequency-dependent, dynamic, staggered spin susceptibility in terms of four experimentally measurable, T=0 parameters. The scaling function contains the full scaling behavior in all regimes for both type-A and -B systems. The crossover behavior of the uniform susceptibility and the specific heat is somewhat more complicated and is also discussed. Explicit computation of the crossover functions is carried out in a large N expansion on a mean-field model. Some new results for the critical properties on the ordered side of the transition are also obtained in a spin-density-wave formalism. The possible relevance of our results to the doped cuprate compounds is briefly discussed.