Quantum Critical Fluctuations Research Articles

Transverse Ising model with multi-impurity

Abstract We study the transverse Ising spin model with spin-1 impurities under the exact solution. We develop a universal method to deal with the multi-impurity problem by introducing a displacement quantity in the wave function and get a recursive formula to simplify the calculation of the partition function. This allows us to rigorously determine the impurity effects for a specific distribution of impurity in the thermodynamic limit. The low temperature behaviors are governed by the interplay between host and impurity excitations, and the quantum critical fluctuations around the critical point of the transverse Ising model are tuned by the transverse field and the concentration of impurity. However the impurity effects are limited, which depends on the host–impurity exchange interaction and the coupling strength of impurities.

Dark Energy Stars and the Cosmic Microwave Background

Cosmological models with a large cosmological constant are unstable due to quantum critical fluctuations at the de Sitter horizon. Due to this instability the space-time in these models will quickly evolve into a Friedmann-like expanding universe containing dark energy stars and radiation. In this paper it is pointed out that this provides a simple explanation for both the observed radiation entropy per gram of dark matter and the level of temperature fluctuations in the cosmic microwave background. A novel prediction is that large dark energy stars provide the seeds for the formation of galaxies.

Emergence of superconductivity in heavy-electron materials.

Although the pairing glue for the attractive quasiparticle interaction responsible for unconventional superconductivity in heavy-electron materials has been identified as the spin fluctuations that arise from their proximity to a magnetic quantum critical point, there has been no model to describe their superconducting transition at temperature Tc that is comparable to that found by Bardeen, Cooper, and Schrieffer (BCS) for conventional superconductors, where phonons provide the pairing glue. Here we propose such a model: a phenomenological BCS-like expression for Tc in heavy-electron materials that is based on a simple model for the effective range and strength of the spin-fluctuation-induced quasiparticle interaction and reflects the unusual properties of the heavy-electron normal state from which superconductivity emerges. We show that it provides a quantitative understanding of the pressure-induced variation of Tc in the "hydrogen atoms" of unconventional superconductivity, CeCoIn5 and CeRhIn5, predicts scaling behavior and a dome-like structure for Tc in all heavy-electron quantum critical superconductors, provides unexpected connections between members of this family, and quantifies their variations in Tc with a single parameter.

Quantum critical fluctuations in the heavy fermion compound Ce(Ni0.935Pd0.065)2Ge2

Electric resistivity, specific heat, magnetic susceptibility, and inelastic neutron scattering experiments were performed on a single crystal of the heavy fermion compound Ce(Ni0.935Pd0.065)2Ge2 in order to study the spin fluctuations near an antiferromagnetic (AF) quantum critical point (QCP). The resistivity and the specific heat coefficient for T ⩽ 1 K exhibit the power law behavior expected for a 3D itinerant AF QCP (ρ(T) ∼ T3/2 and γ(T) ∼ γ0 − bT1/2). However, for 2 ⩽ T ⩽ 10 K, the susceptibility and specific heat vary as log T and the resistivity varies linearly with temperature. Furthermore, despite the fact that the resistivity and specific heat exhibit the non-Fermi liquid behavior expected at a QCP, the correlation length, correlation time, and staggered susceptibility of the spin fluctuations remain finite at low temperature. We suggest that these deviations from the divergent behavior expected for a QCP may result from alloy disorder.

Anomalous critical fields in quantum critical superconductors

Fluctuations around an antiferromagnetic quantum critical point (QCP) are believed to lead to unconventional superconductivity and in some cases to high-temperature superconductivity. However, the exact mechanism by which this occurs remains poorly understood. The iron-pnictide superconductor BaFe2(As1−xPx)2 is perhaps the clearest example to date of a high-temperature quantum critical superconductor, and so it is a particularly suitable system to study how the quantum critical fluctuations affect the superconducting state. Here we show that the proximity of the QCP yields unexpected anomalies in the superconducting critical fields. We find that both the lower and upper critical fields do not follow the behaviour, predicted by conventional theory, resulting from the observed mass enhancement near the QCP. Our results imply that the energy of superconducting vortices is enhanced, possibly due to a microscopic mixing of antiferromagnetism and superconductivity, suggesting that a highly unusual vortex state is realized in quantum critical superconductors.

Non-Fermi-liquid behavior at the onset of incommensurate2kFcharge- or spin-density wave order in two dimensions

We analyze the influence of quantum critical fluctuations on single-particle excitations at the onset of incommensurate $2k_F$ charge or spin density wave order in two-dimensional metals. The case of a single pair of hot spots at high symmetry positions on the Fermi surface needs to be distinguished from the case of two hot spot pairs. We compute the fluctuation propagator and the electronic self-energy perturbatively in leading order. The energy dependence of the single-particle decay rate at the hot spots obeys non-Fermi liquid power laws, with an exponent 2/3 in the case of a single hot spot pair, and exponent one for two hot spot pairs. The prefactors of the linear behavior obtained in the latter case are not particle-hole symmetric.

Quantum critical fluctuations in layered YFe2Al10.

The absence of thermal fluctuations at T = 0 makes it possible to observe the inherently quantum mechanical nature of systems where the competition among correlations leads to different types of collective ground states. Our high precision measurements of the magnetic susceptibility, specific heat, and electrical resistivity in the layered compound YFe2Al10 demonstrate robust field-temperature scaling, evidence that this system is naturally poised without tuning on the verge of ferromagnetic order that occurs exactly at T = 0, where magnetic fields drive the system away from this quantum critical point and restore normal metallic behavior.

Heavy fermions: superconductivity and its relationship to quantum criticality

Anti-ferromagnetic (AF) heavy-fermion (HF) metals frequently exhibit a quantum critical point (QCP), i.e. a continuous disappearance of AF order at zero temperature as a function of a non-thermal control parameter, like pressure. Commonly, unconventional (d-wave) superconductivity (SC) is observed in the vicinity of such a QCP. In this article, the interplay between quantum criticality and SC in HF metals is discussed. The latter can be well described in the framework of the Kondo lattice (KL) model. After introducing HFs and giving an overview on HF superconductivity, the characteristic energy scales in KL systems are addressed. Subsequently, we concentrate on the isostructural compounds CeCu2Si2 and YbRh2Si2. CeCu2Si2 is a prototypical HF superconductor which shows a QCP of itinerant, i.e. three-dimensional (3D) spin-density-wave (SDW) type. Overdamped, nearly quantum-critical SDW fluctuations are concluded from inelastic-neutron-scattering results to be the driving force for SC (Tc ≈ 0.6 K) in this compound. While the QCP in several other Ce-based HF superconductors is presumably of the SDW variety too, it is likely to be of the local, i.e. Kondo destroying type in the pressure-induced superconductor CeRhIn5. YbRh2Si2 has been identified as a prototype HF metal exhibiting such a Kondo breakdown QCP, often named a (T = 0) 4f-orbital selective Mott transition. This compound is not a superconductor at T ˃ 10 mK. Systematically searching for and studying SC near Kondo destroying QCPs may offer a link to unconventional SC in other families of correlated materials, such as the doped Mott insulators in the high-Tc cuprates and some of the organic charge-transfer salts.

Relationship between Superconductivity and Antiferromagnetism in LaFe(As1−xPx)O Revealed by 31P-NMR

We performed $^{31}$P-NMR measurements on LaFe(As$_{1-x}$P$_{x}$)O to investigate the relationship between antiferromagnetism and superconductivity. The antiferromagnetic (AFM) ordering temperature $T_{\rm N}$ and the moment $\mu_{\rm ord}$ are continuously suppressed with increasing P content $x$ and disappear at $x = 0.3$ where bulk superconductivity appears. At this superconducting $x = 0.3$, quantum critical AFM fluctuations are observed, indicative of the intimate relationship between superconductivity and low-energy AFM fluctuations associated with the quantum-critical point in LaFe(As$_{1-x}$P$_{x}$)O. The relationship is similar to those observed in other isovalent-substitution systems, e.g., BaFe$_{2}$(As$_{1-x}$P$_{x}$)$_{2}$ and SrFe$_{2}$(As$_{1-x}$P$_{x}$)$_{2}$, with the "122" structure. Moreover, the AFM order reappears with further P substitution ($x > 0.4$). The variation of the ground state with respect to the P substitution is considered to be linked to the change in the band character of Fe-3$d$ orbitals around the Fermi level.

Strong coupling behavior of the neutron resonance mode in unconventional superconductors

We analyze whether and how the neutron resonance mode in unconventional superconductors is affected by higher order corrections in the coupling between spin excitations and fermionic quasiparticles and find that in general such corrections cannot be ignored. In particular, we find that in two spatial dimensions (d=2) the corrections are of same order as the leading, weak coupling contributions demonstrating that the neutron resonance mode in unconventional superconductors is a strong coupling phenomenon. The origin of this behavior lies in the quantum-critical nature of the low energy spin dynamics in the superconducting state and the feedback of the resonance mode onto the fermionic excitations. While quantum critical fluctuations occur in any dimensionality smaller than the upper critical dimension d_{uc}=3, they can be analyzed in a controlled fashion by means of the \epsilon-expansion (\epsilon =3-d), such that the leading corrections to the resonance mode position are small. Regardless of the strong coupling nature of the resonance mode we show that it emerges only if the phase of the superconducting gap function varies on the Fermi surface, making it a powerful tool to investigate the microscopic structure of the pair condensate.

Superconducting transition temperatures for spin-fluctuation superconductivity: Application to heavy-fermion compounds

The quantum critical Antiferromagnetic (AFM) fluctuation spectra measured by inelastic neutron scattering recently in two heavy fermion superconductors are used together with their other measured properties to calculate their D-wave superconducting transition temperatures $T_{\rm c}$. To this end, the linearized Eliashberg equations for D-wave superconductivity induced by AFM fluctuations are solved in models of fermions with various levels of nesting. The results for the ratio of $T_{\rm c}$ to the characteristic spin-fluctuation energy are well parametrized by a dimensionless coupling constant and the AFM correlation length. Comparing the results with experiments suggests that one may reasonably conclude that superconductivity in these compounds is indeed caused by AFM fluctuations. This conclusion is strengthened by a calculation with the same parameters of the measured coefficient of the normal state quantum-critical resistivity $\propto T^{3/2}$ characteristic of {\it gaussian} AFM quantum-critical fluctuations. The calculations give details of the superconducting coupling as a function of the correlation length and the integrated fluctuation spectra useful in other compounds.

Quantum-limit linkage of ‘strange’ and conventional metal states of high-Tc superconductors

The key to the nature of superconducting pairing in high-Tc cuprates lies in understanding the transition to a conventional behavior when they are heavily doped. By using high magnetic fields, we show that the pseudogapped Tl2Ba2CuO6+x becomes a conventional metal at heavy doping in a quantum phase transition, where the pseudogap boundary develops a thermodynamic divergence corresponding to a zero entropy jump. The critical doping point downshifts with magnetic field in unison with the suppression of Tc. This implies that quantum critical fluctuations and magnetic degrees that destabilize the pseudogap are linked to the superconductivity with high Tc.

Comparison of 2-D Quantum and Thermal Critical Fluctuations of Underdoped Bi2Sr2CaCu2O8+δ with Ultrathin YBa2Cu3O7−δ

This paper addresses two issues. Underdoped Bi-2212 has been shown to exhibit two-dimensional quantum fluctuations, with Tc vs. Ns(0) scaling qualitatively similar to ultrathin YBCO films that are 2-D by construction. The present paper shows that the quantitative difference in scaling is resolved by growing Bi-2212 films that are much smoother than previously used. Also, the absence of 2-D thermal critical behavior from the superfluid density of Bi-2212 films that exhibit 2-D quantum scaling is not likely due to an inhomogeneous Tc smearing out the 2-D thermal transition. Finally, we emphasize that there is as yet no consensus on the physics that drives the T-dependence of Ns at low to intermediate temperatures.

Heavy-fermion superconductivity in the Kondo-lattice system CeCu2Si2

The formation of “heavy fermions” (HFs), i.e., charge carriers with largely enhanced effective mass, in Ce-based Kondo lattice (KL) systems is recalled. In contrast to the disastrous pair-breaking effect of magnetic Ce3+ impurities in certain superconducting Kondo alloys, a periodic dense array of Ce3+ ions is a prerequisite for the generation of HF superconductivity (SC) below Tc = 0.6 K in the KL system CeCu2Si2. Subsequent to this discovery in 1979, more than 35 other Ce-, Pr-, Yb-, U-, Np- and Pu-based HF superconductors (with maximum Tc = 18.5 K) have been found. These materials exhibit a variety of fascinating properties, e.g., even- or odd-parity Cooper pairs, multi-phase diagrams or the coexistence with magnetic order. In particular, the interplay between HF SC and quantum criticality is of high timely interest. CeCu2Si2 has emerged as a prototype system showing a three-dimensional (3D) spindensity-wave (SDW) quantum-critical point. Recent results of inelastic-neutron-scattering experiments on this compound are briefly discussed. They lend strong support to the perception of Cooper pairs mediated by nearly quantum-critical 3D-SDW fluctuations covering a wide frequency range.

Stripe melting and quantum criticality in correlated metals

We study theoretically quantum melting transitions of stripe order in a metallic environment, and the associated reconstruction of the electronic Fermi surface. We show that such quantum phase transitions can be continuous in situations where the stripe melting occurs by proliferating pairs of dislocations in the stripe order parameter without proliferating single dislocations. We develop an intuitive picture of such phases as "Stripe Loop Metals" where the fluctuating stripes form closed loops of arbitrary size at long distances. We obtain a controlled critical theory of a few different continuous quantum melting transitions of stripes in metals . At such a (deconfined) critical point the fluctuations of the stripe order parameter are strongly coupled, yet tractable. They also decouple dynamically from the Fermi-surface. We calculate many universal properties of these quantum critical points. In particular we find that the full Fermi-surface and the associated Landau quasiparticles remain sharply defined at the critical point. We discuss the phenomenon of Fermi surface reconstruction across this transition and the effect of quantum critical stripe fluctuations on the superconducting instability. We study possible relevance of our results to several phenomena in the cuprates.

Evidence of two-dimensional quantum critical behavior in the superfluid density of extremely underdoped Bi2Sr2CaCu2O8+x

Evidence of two-dimensional (2D) quantum critical fluctuations is observed in the superfluid density ns(T) \propto $\lambda$ -2(T) of deeply underdoped Bi2Sr2CaCu2O8+x (Bi-2212) films, indicating that quantum fluctuations play a dominant role in underdoped cuprates in general. 2D fluctuations are expressed by the linear scaling, Tc \propto ns(0). 2D scaling in Bi-2212 contrasts with 3D scaling seen in the much less anisotropic YBa2Cu3O7-x. Quantum critical fluctuations could also account for the absence of thermal critical behavior in \lambda^{-2}(T) of strongly underdoped Bi-2212 samples, Tc < 48 K.

Order parameter fluctuations at a buried quantum critical point

Quantum criticality is a central concept in condensed matter physics, but the direct observation of quantum critical fluctuations has remained elusive. Here we present an X-ray diffraction study of the charge density wave (CDW) in 2H-NbSe(2) at high pressure and low temperature, where we observe a broad regime of order parameter fluctuations that are controlled by proximity to a quantum critical point. X-rays can track the CDW despite the fact that the quantum critical regime is shrouded inside a superconducting phase; and in contrast to transport probes, allow direct measurement of the critical fluctuations of the charge order. Concurrent measurements of the crystal lattice point to a critical transition that is continuous in nature. Our results confirm the long-standing expectations of enhanced quantum fluctuations in low-dimensional systems, and may help to constrain theories of the quantum critical Fermi surface.

Thermal and electrical transport across a magnetic quantum critical point

A quantum critical point (QCP) arises when a continuous transition between competing phases occurs at zero temperature. Collective excitations at magnetic QCPs give rise to metallic properties that strongly deviate from the expectations of Landau's Fermi-liquid description, which is the standard theory of electron correlations in metals. Central to this theory is the notion of quasiparticles, electronic excitations that possess the quantum numbers of the non-interacting electrons. Here we report measurements of thermal and electrical transport across the field-induced magnetic QCP in the heavy-fermion compound YbRh(2)Si(2) (refs 2, 3). We show that the ratio of the thermal to electrical conductivities at the zero-temperature limit obeys the Wiedemann-Franz law for magnetic fields above the critical field at which the QCP is attained. This is also expected for magnetic fields below the critical field, where weak antiferromagnetic order and a Fermi-liquid phase form below 0.07 K (at zero field). At the critical field, however, the low-temperature electrical conductivity exceeds the thermal conductivity by about 10 per cent, suggestive of a non-Fermi-liquid ground state. This apparent violation of the Wiedemann-Franz law provides evidence for an unconventional type of QCP at which the fundamental concept of Landau quasiparticles no longer holds. These results imply that Landau quasiparticles break up, and that the origin of this disintegration is inelastic scattering associated with electronic quantum critical fluctuations--these insights could be relevant to understanding other deviations from Fermi-liquid behaviour frequently observed in various classes of correlated materials.

Thermodynamics of the Classical Planar Ferromagnet Close to the Zero-Temperature Critical Point: A Many-Body Approach

We explore the low-temperature thermodynamic properties and crossovers of ad-dimensional classical planar Heisenberg ferromagnet in a longitudinal magnetic field close to its field-induced zero-temperature critical point by employing the two-time Green’s function formalism in classical statistical mechanics. By means of a classical Callen-like method for the magnetization and the Tyablikov-like decoupling procedure, we obtain, for anyd, a low-temperature critical scenario which is quite similar to the one found for the quantum counterpart. Remarkably, ford&gt;2the discrimination between the two cases is found to be related to the different values of the shift exponent which governs the behavior of the critical line in the vicinity of the zero-temperature critical point. The observation of different values of the shift-exponent and of the related critical exponents along thermodynamic paths within the typical V-shaped region in the phase diagram may be interpreted as a signature of emerging quantum critical fluctuations.

Superconductivity in the cuprates: Deduction of mechanism for d-wave pairing through analysis of ARPES

In the Eliashberg integral equations for d-wave superconductivity, two different functions (α2F)n(ω, θ) and (α2F)p,d(ω) determine, respectively, the “normal” self-energy and the “pairing” self-energy. ω is the frequency of fluctuations scattering the fermions whose momentum is near the Fermi-surface and makes an angle θ to a chosen axis. We present a quantitative analysis of the high-resolution laser based Angle Resolved Photoemission Spectroscopy (ARPES) data on a slightly under doped cuprate compound Bi2212 and use the Eliashberg equations to deduce the ω and θ dependence of (α2F)n(ω, θ) for T just above Tc and below Tc. Besides its detailed ω dependence, we find the remarkable result that this function is nearly independent of θ between the (π; π)-direction and 25 degrees from it, except for the dependence of the cut-off energy on θ. Assuming that the same fluctuations determine both the normal and the pairing self-energy, we ask what theories give the function (α2F)p,d(ω) required for the d-wave pairing instability at high temperatures as well as the deduced (α2F)n(θ, ω). We show that the deduced (α2F)n(θ, ω) can only be obtained from antiferromagnetic (AFM) fluctuations if their correlation length is smaller than a lattice constant. Using (α2F)p,d(ω) consistent with such a correlation length and the symmetry of matrix-elements scattering fermions by AFM fluctuations, we calculate Tc and show that AFM fluctuations are excluded as the pairing mechanism for d-wave superconductivity in cuprates. We also consider the quantumcritical fluctuations derived microscopically as the fluctuations of the observed loop-current order discovered in the under-doped cuprates, and which lead to the marginal Fermi-liquid properties in the normal state. We show that their frequency dependence and the momentum dependence of their matrix-elements to scatter fermions are consistent with the θ and ω dependence of the deduced (α2F)n(ω, θ). The pairing kernel (α2F)p,d(ω) calculated using the experimental values in the Eliashberg equation gives d-wave instability at Tc comparable to the experiments.