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Physical review. B, Condensed matter

Spin dynamics of a two-dimensional metal in a superconducting state: Application to the high- Tc cuprates

F. P. Onufrieva, Pierre M. Pfeuty

Published Jan 11, 2002• 81 citations
Publisher

Abstract

We show that a two-dimensional (2D) metal in a superconducting (SC) state with wave-vector-dependent gap behaves as a quantum-spin liquid characterized by a collective mode of resonance type. Formally the mode exists whatever is a form of Fermi surface (FS). However, when the latter is spherical the effect is rather formal: any factor leading to an even weak electron scattering, like finite T, impurity, etc., easily kills the mode. On the contrary, for the type of FS observed in the high-${T}_{c}$ cuprates (characterized by a proximity to the saddle-point wave vectors) the resonance mode exists in a true sense being well separated from the electron-hole continuum and well defined. The effect is due to the effective 1D dimension in a proximity of electronic topological transition quantum critical point. The physical nature of the collective mode varies progressively between two possibilities. In the case of a weak interaction, the resonance mode is a typical effect of a two-particle bound state that appears slightly below the bottom of the two-particle continuum. The effect exists even in the case of infinitesimally weak interaction (of relevant sign). In the case of a strong or intermediate interaction, it is a critical mode existing as a precursor of spin-density wave (SDW) instability. The gap of the collective mode is not related directly to the value of the SC gap (being, however, limited by the latter from above) but rather to a proximity to SDW instability. We predict also a second collective mode lying inside the two-particle continuum. The existence of these two modes and the particular form of their dispersion in the case corresponding to the underdoped cuprates ${(d}_{{x}^{2}\ensuremath{\sim}{y}^{2}}$ symmetry of SC gap, exchange interaction of antiferromagnetic sign, etc.) allows one to understand the main features observed experimentally in the underdoped cuprates, namely the so-called ``41 meV resonance peak'' and ``24 meV incommensurability'' and their two-dimensional wave vector dependences, the doping evolution of the spin dynamics and many other features. The theory also predicts new features to be tested experimentally.

DOI
10.1103/physrevb.65.054515

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We show that a two-dimensional (2D) metal in a superconducting (SC) state with wave-vector-dependent gap behaves as a quantum-spin liquid characterized by a collective mode of resonance type. Formally the mode exists whatever is a form of Fermi surface (FS). However, when the latter is spherical the effect is rather formal: any factor leading to an even weak electron scattering, like finite T, impurity, etc., easily kills the mode. On the contrary, for the type of FS observed in the high-Tc{T}_{c}Tc​ cuprates (characterized by a proximity to the saddle-point wave vectors) the resonance mode exists in a true sense being well separated from the electron-hole continuum and well defined. The effect is due to the effective 1D dimension in a proximity of electronic topological transition quantum critical point. The physical nature of the collective mode varies progressively between two possibilities. In the case of a weak interaction, the resonance mode is a typical effect of a two-particle bound state that appears slightly below the bottom of the two-particle continuum. The effect exists even in the case of infinitesimally weak interaction (of relevant sign). In the case of a strong or intermediate interaction, it is a critical mode existing as a precursor of spin-density wave (SDW) instability. The gap of the collective mode is not related directly to the value of the SC gap (being, however, limited by the latter from above) but rather to a proximity to SDW instability. We predict also a second collective mode lying inside the two-particle continuum. The existence of these two modes and the particular form of their dispersion in the case corresponding to the underdoped cuprates (dx2\ensuremath∼y2{(d}_{{x}^{2}\ensuremath{\sim}{y}^{2}}(dx2\ensuremath∼y2​ symmetry of SC gap, exchange interaction of antiferromagnetic sign, etc.) allows one to understand the main features observed experimentally in the underdoped cuprates, namely the so-called 41 meV resonance peak'' and 24 meV incommensurability'' and their two-dimensional wave vector dependences, the doping evolution of the spin dynamics and many other features. The theory also predicts new features to be tested experimentally.
We have discovered a new example of quantum spin liquid which is a superconducting (SC) phase in 2D electron system close to electronic topological transition. As a quantum spin liquid in low dimensional localized spin systems it is characterized by a resonance spin mode and a well separated two-particle continuum. Application of the theory to the high-T_c cuprates allows to shed light on many observed in the underdoped regime features such as the so called 40 meV resonance peak, incommensurability at lower ω, spin gap etc.
A simple interlayer pair tunneling is solved exactly. We find that in the normal state spin-1/2 particle and hole excitations are gapped. But the state is an unusual metal, characterized by novel fermionic spin zero and charge +2e and -2e excitations that exist about their own Fermi surfaces. The model is consistent with a number of properties of the underdoped cuprates. At finite temperature the Fermi surface appears partially gapped, and at high temperatures the spin gap disappears completely. Superconductivity is induced by an additional intralyer interaction which opens a gap in the charge-fermion spectrum. The symmetry of this charge gap (and hence that of the order parameter) is igenerally different from the symmetry of one-electron or - hole gap. The former can be a d wave, while the latter is more complex.
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 describe two-dimensional models with a metallic Fermi surface which display quantum phase transitions controlled by strongly interacting critical field theories below their upper critical dimension. The primary examples involve transitions with a topological order parameter associated with dislocations in collinear spin-density-wave (``stripe'') correlations: the suppression of dislocations leads to a fractionalization of spin and charge collective modes, and this transition has been proposed as a candidate for the cuprates near optimal doping. The coupling between the order parameter and long-wavelength volume and shape deformations of the Fermi surface is analyzed by the renormalization group, and a runaway flow to a nonperturbative regime is found in most cases. A phenomenological scaling analysis of simple observable properties of possible second-order quantum critical points is presented, with results quite similar to those near quantum spin-glass transitions and to phenomenological forms proposed by Schr\"oder et al. [Nature (London) 407,407,407, 351 (2000)].
One of the central issues in the study of high-temperature superconducting cuprates which are composed of two-dimensional (2D) CuO 2 planes is whether the 2D systems with strong electron correlation behave as a Fermi liquid or a non-Fermi-liquid-like one-dimensional (1D) system with electron correlation. In this article, we start with the detailed examination of the electronic structure in cuprates and study theoretically the spin and charge dynamics in 1D and 2D cuprates. The theoretical background of spin-charge separation in the 1D model systems including the Hubbard and t - J models is presented. The first direct observation of collective modes of spin and charge excitations in a 1D cuprate, which are called spinons and holons respectively, in angle-resolved photoemission spectroscopy (ARPES) experiments is reviewed in the light of the theoretical results based on the numerically exact-diagonalization method. The charge and spin dynamics in 1D insulating cuprates is also discussed in connection with the spin-charge separation. The arguments are extended to the 2D cuprates, and the unique aspects of the electronic properties of high-temperature superconductors are discussed. Special emphasis is placed on the d-wave-like excitations in insulating 2D cuprates observed in ARPES experiments. We explain how the excitations are caused by the spin-charge separation. The charge stripes observed in the underdoped cuprates are examined in connection with spin-charge separation in real space.
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.