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arXiv:cond-mat/9411066 · Physical review. B, Condensed matter

Crossover and scaling in a nearly antiferromagnetic Fermi liquid in two dimensions

Subir Sachdev A, Andrey V. Chubukov, Alexander Sokol

Published Jun 1, 1995• 133 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.51.14874

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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.
We consider a two-dimensional Fermi liquid in the vicinity of a spin-density-wave transition to a phase with commensurate antiferromagnetic long-range order. We assume that near the transition, the Fermi surface is large and crosses the magnetic Brillouin zone boundary. We show that under these conditions, the self-energy corrections to the dynamical spin susceptibility, \ensuremath{\chi}(q,\ensuremath{\omega}), and to the quasiparticle spectral function, A(k,\ensuremath{\omega}), are divergent near the transition. We identify and sum the series of most singular diagrams, and obtain a solution for \ensuremath{\chi}(q,\ensuremath{\omega}) and an approximate solution for A(k,\ensuremath{\omega}). We show that (i) A(k) at a given, small \ensuremath{\omega} has an extra peak at k=kF{\mathit{k}}_{\mathit{F}}kF​+\ensuremath{\pi} (shadow band), and (ii) the dispersion near the crossing points is much flatter than for free electrons. The relevance of these results to recent photoemission experiments in YBa2{\mathrm{YBa}}_{2}YBa2​Cu3{\mathrm{Cu}}_{3}Cu3​${\mathrm{O}}_{7\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}$ and Bi2{\mathrm{Bi}}_{2}Bi2​Sr2{\mathrm{Sr}}_{2}Sr2​CaCu2{\mathrm{CaCu}}_{2}CaCu2​O8{\mathrm{O}}_{8}O8​ systems is discussed.
Quantum critical points exist at zero temperature, yet, experimentally their influence seems to extend over a large part of the phase diagram of systems such as heavy-fermion compounds and high-temperature superconductors. Theoretically, however, it is generally not known over what range of parameters the physics is governed by the quantum critical point. We answer this question for the spin-density wave to Fermi-liquid quantum critical point in the two-dimensional Hubbard model. This problem is in the d =2, z =2 universality class. We use the two-particle self-consistent approach, which is accurate from weak to intermediate coupling, and whose critical behavior is the same as for the self-consistent-renormalized approach of Moriya. Despite the presence of logarithmic corrections, numerical results demonstrate that quantum critical scaling for the static magnetic susceptibility can extend up to very high temperatures but that the commensurate to incommensurate crossover leads to deviations to scaling.
The antiferromagnetic spin correlation function SQ{S}_{\mathbf{Q}}SQ​, the staggered spin susceptibility ${\ensuremath{\chi}}_{\mathbf{Q}}$, and the energy scale ${\ensuremath{\omega}}_{FL}={S}_{\mathbf{Q}}∕{\ensuremath{\chi}}_{\mathbf{Q}}$ are studied numerically within the $t\text{\ensuremath{-}}J$ model and the Hubbard model, as relevant to cuprates. It is shown that ${\ensuremath{\omega}}_{FL}$, related to the onset of the non-Fermi-liquid spin response at $T>{\ensuremath{\omega}}_{FL}$, is very low in the regime below the ``optimum'' hole doping ${c}_{h}<{c}_{h}^{*}\ensuremath{\sim}0.16$, while it shows a steep increase in the overdoped regime. A quantitative analysis of the NMR spin-spin relaxation rate 1∕T2G1∕{T}_{2G}1∕T2G​ for various cuprates reveals a similar behavior, indicating a pronounced crossover between a Fermi-liquid and a non-Fermi-liquid behavior as a function of doping.
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/T1/T1/T, namely, z=1z=1z=1 instead of the naive z=2z=2z=2, the $(\ensuremath{\pi},\ensuremath{\pi})$ static spin susceptibility $\ensuremath{\chi}$ like 1/T1/\sqrt{T}1/T​, and the imaginary part of the self-energy at the hot spots like T3/2{T}^{3/2}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.
We report on numerically exact determinantal quantum Monte Carlo simulations of the onset of spin-density-wave (SDW) order in itinerant electron systems captured by a sign-problem-free two-dimensional lattice model. Extensive measurements of the SDW correlations in the vicinity of the phase transition reveal that the critical dynamics of the bosonic order parameter are well described by a dynamical critical exponent z=2z=2z=2, consistent with Hertz-Millis theory, but are found to follow a finite-temperature dependence that does not fit the predicted behavior of the same theory. The presence of critical SDW fluctuations is found to have a strong impact on the fermionic quasiparticles, giving rise to a dome-shaped superconducting phase near the quantum critical point. In the superconducting state we find a gap function that has an opposite sign between the two bands of the model and is nearly constant along the Fermi surface of each band. Above the superconducting Tc{T}_{c}Tc​, our numerical simulations reveal a nearly temperature and frequency independent self-energy causing a strong suppression of the low-energy quasiparticle weight in the vicinity of the hot spots on the Fermi surface. This indicates a clear breakdown of Fermi liquid theory around these points.
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.