Quantum Critical Fluctuations Research Articles

Superconductivity mediated by quantum critical antiferromagnetic fluctuations: The rise and fall of hot spots

In several unconventional superconductors, the highest superconducting transition temperature $T_{c}$ is found in a region of the phase diagram where the antiferromagnetic transition temperature extrapolates to zero, signaling a putative quantum critical point. The elucidation of the interplay between these two phenomena - high-$T_{c}$ superconductivity and magnetic quantum criticality - remains an important piece of the complex puzzle of unconventional superconductivity. In this paper, we combine sign-problem-free Quantum Monte Carlo simulations and field-theoretical analytical calculations to unveil the microscopic mechanism responsible for the superconducting instability of a general low-energy model, called spin-fermion model. In this approach, low-energy electronic states interact with each other via the exchange of quantum critical magnetic fluctuations. We find that even in the regime of moderately strong interactions, both the superconducting transition temperature and the pairing susceptibility are governed not by the properties of the entire Fermi surface, but instead by the properties of small portions of the Fermi surface called hot spots. Moreover, $T_{c}$ increases with increasing interaction strength, until it starts to saturate at the crossover from hot-spots dominated to Fermi-surface dominated pairing. Our work provides not only invaluable insights into the system parameters that most strongly affect $T_{c}$, but also important benchmarks to assess the origin of superconductivity in both microscopic models and actual materials.

Multidimensional entropy landscape of quantum criticality

The Third Law of Thermodynamics states that the entropy of any system in equilibrium has to vanish at absolute zero temperature. At nonzero temperatures, on the other hand, matter is expected to accumulate entropy near a quantum critical point (QCP), where it undergoes a continuous transition from one ground state to another. Here, we determine, based on general thermodynamic principles, the spatial-dimensional profile of the entropy S near a QCP and its steepest descent in the corresponding multidimensional stress space. We demonstrate this approach for the canonical quantum critical compound CeCu6-xAux near its onset of antiferromagnetic order. We are able to link the directional stress dependence of S to the previously determined geometry of quantum critical fluctuations. Our demonstration of the multidimensional entropy landscape provides the foundation to understand how quantum criticality nucleates novel phases such as high-temperature superconductivity.

Excitonic pairing and insulating transition in two-dimensional semi-Dirac semimetals

A sufficiently strong long-range Coulomb interaction can induce excitonic pairing in gapless Dirac semimetals, which generates a finite gap and drives semimetal-insulator quantum phase transition. This phenomenon is in close analogy to dynamical chiral symmetry breaking in high energy physics. In most realistic Dirac semimetals, including suspended graphene, Coulomb interaction is too weak to open an excitonic gap. The Coulomb interaction plays a more important role at low energies in a two-dimensional semi-Dirac semimetal, in which the fermion spectrum is linear in one component of momenta and quadratic in the other, than a Dirac semimetal, and indeed leads to breakdown of Fermi liquid theory. We study dynamical excitonic gap generation in a two-dimensional semi-Dirac semimetal by solving the Dyson-Schwinger equation, and show that a moderately strong Coulomb interaction suffices to induce excitonic pairing. Additional short-range four-fermion coupling tends to promote excitonic pairing. Among the available semi-Dirac semimetals, we find that TiO$_{2}$/VO$_{2}$ nanostructure provides a promising candidate for the realization of excitonic insulator. We also apply the renormalziation group method to analyze the strong coupling between the massless semi-Dirac fermions and the quantum critical fluctuation of excitonic order parameter at the semimetal-insulator quantum critical point, and reveal non-Fermi liquid behaviors of semi-Dirac fermions.

Phase diagram and correlation functions of the two-dimensional dissipative quantum XY model

The two-dimensional dissipative quantum XY model is applicable to the quantum-critical properties of diverse experimental systems, ranging from the superconductor to insulator transitions, ferromagnetic and antiferromagnetic transitions in metals, to the loop-current order transition in the cuprates. We solve the re-expression of this model in terms of topological excitations: vortices and a variety of instantons, by renormalization group methods. The calculations explain the extraordinary properties of the model discovered in Monte-Carlo calculations: the separability of the quantum critical fluctuations (QCF) in space and time, the spatial correlation length proportional to logarithm of the temporal correlation length near the transition from disordered to the fully ordered state, and the occurrence of a phase with spatial order without temporal order. They are intimately related to the flow of the metric of time in relation to the metric of space, i.e. of the dynamical critical exponent $z$. These properties appear to be essential in understanding the strange metallic phase found in a variety of quantum-critical transitions as well as the accompanying high temperature superconductivity.

Nematic Quantum Critical Fluctuations in BaFe_{2-x}Ni_{x}As_{2}.

We have systematically studied the nematic fluctuations in the electron-doped iron-based superconductor BaFe_{2-x}Ni_{x}As_{2} by measuring the in-plane resistance change under uniaxial pressure. While the nematic quantum critical point can be identified through the measurements along the (110) direction, as studied previously, quantum and thermal critical fluctuations cannot be distinguished due to similar Curie-Weiss-like behaviors. Here we find that a sizable pressure-dependent resistivity along the (100) direction is present in all doping levels, which is against the simple picture of an Ising-type nematic model. The signal along the (100) direction becomes maximum at optimal doping, suggesting that it is associated with nematic quantum critical fluctuations. Our results indicate that thermal fluctuations from striped antiferromagnetic order dominate the underdoped regime along the (110) direction. We argue that either there is a strong coupling between the quantum critical fluctuations and the fermions, or more exotically, a higher symmetry may be present around optimal doping.

Evolution of quasiparticle excitations with enhanced electron correlations in superconductingAFe2As2(A=K, Rb, and Cs)

In the heavily hole-doped iron-based superconductors $A$Fe$_2$As$_2$ ($A=$ K, Rb, and Cs), the electron effective mass increases rapidly with alkali-ion radius. To study how the mass enhancement affects the superconducting state, we measure the London penetration depth $\lambda(T)$ in clean crystals of $A$Fe$_2$As$_2$ down to low temperature $T\sim0.1$ K. In all systems, the superfluid stiffness $\rho_s(T)=\lambda^2(0)/\lambda^2(T)$ can be approximated by a power-law $T$ dependence at low temperatures, indicating the robustness of strong momentum anisotropy in the superconducting gap $\Delta(k)$. The power $\alpha$ increases from $\sim1$ with mass enhancement and approaches an unconventional exponent $\alpha\sim 1.5$ in the heaviest CsFe$_2$As$_2$. This appears to be a hallmark of superconductors near antiferromagnetic quantum critical points, where the quasiparticles excited across the anisotropic $\Delta(k)$ are significantly influenced by the momentum dependence of quantum critical fluctuations.

Unconventional non-Fermi liquid state caused by nematic criticality in cuprates

At the nematic quantum critical point that exists in the -wave superconducting dome of cuprates, the massless nodal fermions interact strongly with the quantum critical fluctuation of nematic order. We study this problem by means of the renormalization group approach and show that, the fermion damping rate vanishes more rapidly than the energy ω and the quasiparticle residue in the limit . The nodal fermions thus constitute an unconventional non-Fermi liquid that represents an even weaker violation of Fermi liquid theory than a marginal Fermi liquid. We also investigate the interplay of quantum nematic critical fluctuation and gauge-potential-like disorder, and find that the effective disorder strength flows to the strong coupling regime at low energies. Therefore, even an arbitrarily weak disorder can drive the system to become a disorder controlled diffusive state. Based on these theoretical results, we are able to understand a number of interesting experimental facts observed in curpate superconductors.

Topological basis for understanding the behavior of the heavy-fermion metalβ−YbAlB4under application of magnetic field and pressure

Informative recent measurements on the heavy-fermion metal $\rm \beta-YbAlB_4$ performed with applied magnetic field and pressure as control parameters are analyzed with the goal of establishing a sound theoretical explanation for the inferred scaling laws and non-Fermi-liquid (NFL) behavior, which demonstrate some unexpected features. Most notably, the robustness of the NFL behavior of the thermodynamic properties and of the anomalous $T^{3/2}$ temperature dependence of the electrical resistivity under applied pressure $P$ in zero magnetic field $B$ is at variance with the fragility of the NFL phase under application of a field. We show that a consistent topological basis for this combination of observations, as well as the empirical scaling laws, may be found within fermion-condensation theory in the emergence and destruction of a flat band, and explain that the paramagnetic NFL phase takes place without magnetic criticality, thus not from quantum critical fluctuations. Schematic $T-B$ and $T-P$ phase diagrams are presented to illuminate this scenario.

Towards ferromagnetic quantum criticality inFeGa3−xGex:Ga71NQR as a zero-field microscopic probe

$^{71}\mathrm{Ga}$ NQR, magnetization, and specific-heat measurements have been performed on polycrystalline Ge-doped ${\mathrm{FeGa}}_{3}$ samples. A crossover from an insulator to a correlated local moment metal in the low-doping regime and the evolution of itinerant ferromagnet upon further doping is found. For the nearly critical concentration at the threshold of ferromagnetic order, ${x}_{C}=0.15, ^{71}(1/{T}_{1}T)$ exhibits a pronounced ${T}^{\ensuremath{-}4/3}$ power law over two orders of magnitude in temperature, which indicates three-dimensional quantum critical ferromagnetic fluctuations. Furthermore, for the ordered $x=0.2$ sample (${T}_{C}\ensuremath{\approx}6$ K), $^{71}(1/{T}_{1}T)$ could be fitted well in the frame of Moriya's self-consistent renormalization theory for weakly ferromagnetic systems with $1/{T}_{1}T\ensuremath{\sim}\ensuremath{\chi}$. In contrast to this, the low-doping regime nicely displays local moment behavior where $1/{T}_{1}T\ensuremath{\sim}{\ensuremath{\chi}}^{2}$ is valid. For $T\ensuremath{\rightarrow}0$, the Sommerfeld ratio $\ensuremath{\gamma}=(C/T)$ is enhanced $(70\phantom{\rule{0.16em}{0ex}}\mathrm{mJ}/\mathrm{mole}\phantom{\rule{0.28em}{0ex}}{\mathrm{K}}^{2}$ for $x=0.1)$, which indicates the formation of heavy $3d$ electrons.

Quantum tricritical fluctuations driving mass enhancement and reentrant superconductivity in URhGe

The field-induced reentrant superconductivity (RSC) discovered near a quantum critical point (QCP) in a ferromagnetic superconductor URhGe highlights the close interplay between superconductivity and magnetism. While the origin of the RSC is broadly thought to be associated with quantum critical fluctuations, their exact nature had not been well identified. Here we review our recent 59Co NMR study in a single crystal of URh0.9Co0.1Ge. Our measurements of the NMR spin-spin relaxation reveal a divergence of electronic spin fluctuations in the vicinity of the field-induced QCP at HR ≈ 13 T. The fluctuations observed are characteristic of a tricritical point, followed by a phase bifurcation toward quantum wing-critical points. We show that these tricritical fluctuations enhance the effective mass of the conduction electrons and, further, drive the RSC near the HR.

Higgs mechanism, phase transitions, and anomalous Hall effect in three-dimensional topological superconductors

We demonstrate that the Higgs mechanism in three-dimensional topological superconductors exhibits unique features with experimentally observable consequences. The Higgs model we discuss has two superconducting components and an axion-like magnetoelectric term with the phase difference of the superconducting order parameters playing the role of the axion field. Due to this additional term, quantum electromagnetic and phase fluctuations lead to a robust topologically non-trivial state that holds also in the presence of interactions. In this sense, we show that the renormalization flow of the topologically nontrivial phase cannot be continuously deformed into a topologically non-trivial one. One consequence of our analysis of quantum critical fluctuations, is the possibility of having a first-order phase transition in the bulk and a second-order phase transition on the surface. We also explore another consequence of the axionic Higgs electrodynamics, namely, the anomalous Hall effect. In the low frequency London regime an anomalous Hall effect is induced in the presence of an applied electric field parallel to the surface. This anomalous Hall current is induced by a Lorentz-like force arising from the axion term, and it involves the relative superfluid velocity of the superconducting components. The anomalous Hall current has a negative sign, a situation reminiscent of, but quite distinct in physical origin from the anomalous Hall effect observed in high-$T_c$ superconductors. In contrast to the latter, the anomalous Hall effect in topological superconductors is non-dissipative and occurs in the absence of vortices.

Quantum critical scaling and superconductivity in heavy electron materials

We use the two fluid model to determine the conditions under which the nuclear spin-lattice lattice relaxation rate, $T_1$, of candidate heavy quantum critical superconductors can exhibit scaling behavior and find that it can occur if and only if their "hidden" quantum critical spin fluctuations give rise to a temperature-independent intrinsic heavy electron spin-lattice relaxation rate. The resulting scaling of $T_1$ with the strength of the heavy electron component and the coherence temperature, $T^*$, provides a simple test for their presence at pressures at which the superconducting transition temperature, $T_c$, is maximum and is proportional to $T^*$. These findings support the previously noted partial scaling of the spin-lattice relaxation rate with $T_c$ in a number of important heavy electron materials and provide additional evidence that in these materials their optimal superconductivity originates in the quantum critical spin fluctuations associated with a nearby phase transition from partially localized to fully itinerant quasiparticles.

Quantum critical response function in quasi-two-dimensional itinerant antiferromagnets

We re-examine the experimental results for the magnetic response function $\chi''({\bf q}, E, T)$, for ${\bf q}$ around the anti-ferromagnetic vectors ${\bf Q}$, in the quantum-critical region, obtained by inelastic neutron scattering, on an Fe-based superconductor, and on a heavy Fermion compound. The motivation is to compare the results with a recent theory, which shows that the fluctuations in a generic anti-ferromagnetic model for itinerant fermions map to those in the universality class of the dissipative quantum-XY model. The quantum-critical fluctuations in this model, in a range of parameters, are given by the correlations of spatial and of temporal topological defects. The theory predicts a $\chi''({\bf q}, E, T)$ (i) which is a separable function of $({\bf q -Q})$ and of ($E$,$T$), (ii) at crticality, the energy dependent part is $\propto \tanh (E/2T)$ below a cut-off energy, (iii) the correlation time departs from its infinite value at criticality on the disordered side by an essential singularity, and (iv) the correlation length depends logarithmically on the correlation time, so that the dynamical critical exponent $z$ is $\infty$ . The limited existing experimental results are found to be consistent with the first two unusual predictions from which the linear dependence of the resistivity on T and the $T \ln T$ dependence of the entropy also follow. More experiments are suggested, especially to test the theory of variations on the correlation time and length on the departure from criticality.

Interplay between density and superconducting quantum critical fluctuations

We consider the case of a density-driven metal-superconductor transition in the proximity of an electronic phase separation. In particular, we investigate the interplay between superconducting fluctuations and density fluctuations, which become quantum critical when the electronic phase separation vanishes at zero temperature into a quantum critical point. In this situation, the critical dynamical density fluctuations strongly affect the dynamics of the Cooper-pair fluctuations, which acquire a more singular character with a z = 3 dynamical critical index. This gives rise to a scenario that possibly rules the disappearance of superconductivity when the electron density is reduced by electrostatic gating at the LaAlO3/SrTiO3 interface.

Kondo destruction in heavy fermion quantum criticality and the photoemission spectrum of YbRh2Si2

Heavy fermion metals provide a prototype setting to study quantum criticality. Experimentally, quantum critical points have been identified and studied in a growing list of heavy fermion compounds. Theoretically, Kondo destruction has provided a means to characterize a class of unconventional quantum critical points that goes beyond the Landau framework of order-parameter fluctuations. Among the prominent evidence for such local quantum criticality have been measurements in YbRh2Si2. A rapid crossover is observed at finite temperatures in the isothermal field dependence of the Hall coefficient and other transport and thermodynamic quantities, which specifies a T*B line in the temperature (T)–magnetic field (B) phase diagram. Here, we discuss what happens when temperature is raised, by analyzing the ratio of the crossover width to the crossover position. With this ratio approaching unity at T≳0.5K, YbRh2Si2 at zero magnetic field belongs to the quantum-critical fluctuation regime, where the single-particle spectral function has significant weight at both the small and large Fermi surfaces. This implies that, in this temperature range, any measurements sensitive to the Fermi surface will also see a significant spectral weight at the large Fermi surface. The angle-resolved photoemission spectroscopy experiments recently reported for YbRh2Si2 at T>1K are consistent with this expectation, and therefore support the association of the T*B line with the physics of Kondo destruction.

HEAVY FERMIONS. Strange metal without magnetic criticality.

A fundamental challenge to our current understanding of metals is the observation of qualitative departures from Fermi liquid behavior. The standard view attributes such non-Fermi liquid phenomena to the scattering of electrons off quantum critical fluctuations of an underlying order parameter. Although the possibility of non-Fermi liquid behavior isolated from the border of magnetism has long been speculated, no experimental confirmation has been made. Here, we report on the observation of a strange metal region away from a magnetic instability in an ultrapure single crystal. In particular, we show that the heavy-fermion superconductor β-YbAlB4 forms a possible phase with strange metallic behavior across an extensive pressure regime, distinctly separated from a high-pressure magnetic quantum phase transition by a Fermi liquid phase.

Anomalous scaling of the penetration depth in nodal superconductors

Recent findings of anomalous super-linear scaling of low temperature ($T$) penetration depth (PD) in several nodal superconductors near putative quantum critical points suggest that the low temperature PD can be a useful probe of quantum critical fluctuations in a superconductor. On the other hand, cuprates which are poster child nodal superconductors have not shown any such anomalous scaling of PD, despite growing evidence of quantum critical points. Then it is natural to ask when and how can quantum critical fluctuations cause anomalous scaling of PD? Carrying out the renormalization group calculation for the problem of two dimensional superconductors with point nodes, we show that quantum critical fluctuations associated with point group symmetry reduction result in non-universal logarithmic corrections to the $T$-dependence of the PD. The resulting apparent power law depends on the bare velocity anisotropy ratio. We then compare our results to data sets from two distinct nodal superconductors: YBa$_2$Cu$_3$O$_{6.95}$ and CeCoIn$_5$. Considering all symmetry-lowering possibilities of the point group of interest, $C_{4v}$, we find our results to be remarkably consistent with YBa$_2$Cu$_3$O$_{6.95}$ being near vertical nematic QCP, and CeCoIn$_5$ being near diagonal nematic QCP. Our results motivate search for diagonal nematic fluctuations in CeCoIn$_5$.

Signatures of nematic quantum critical fluctuations in the Raman spectra of lightly doped cuprates

We consider the lightly doped cuprates Y$_{0.97}$Ca$_{0.03}$BaCuO$_{6.05}$ and La$_{2-x}$Sr$_x$CuO$_4$ (with $x=0.02$,0.04), where the presence of a fluctuating nematic state has often been proposed as a precursor of the stripe (or, more generically, charge-density wave) phase, which sets in at higher doping. We phenomenologically assume a quantum critical character for the longitudinal and transverse nematic, and for the charge-ordering fluctuations, and investigate the effects of these fluctuations in Raman spectra. We find that the longitudinal nematic fluctuations peaked at zero transferred momentum account well for the anomalous Raman absorption observed in these systems in the $B_{2g}$ channel, while the absence of such effect in the $B_{1g}$ channel may be due to the overall suppression of Raman response at low frequencies, associated with the pseudogap. While in Y$_{0.97}$Ca$_{0.03}$BaCuO$_{6.05}$ the low-frequency lineshape is fully accounted by longitudinal nematic collective modes alone, in La$_{2-x}$Sr$_x$CuO$_4$ also charge-ordering modes with finite characteristic wavevector are needed to reproduce the shoulders observed in the Raman response. This different involvement of the nearly critical modes in the two materials suggests a different evolution of the nematic state at very low doping into the nearly charge-ordered state at higher doping.

Criticalities in the itinerant ferromagnetUGe2

We provide a microscopic description of the magnetic properties of UGe$_2$ and in particular, of its both classical and quantum critical behavior. Namely, we account for all the critical points: the critical ending point (CEP) at the metamagnetic phase transition, the tricritical point, and the quantum critical end point at the ferromagnetic to paramagnetic phase transition. Their position agrees quantitatively with experiment. Additionally, we predict that the metamagnetic CEP can be traced down to zero temperature and becomes quantum critical point by a small decrease of both the total electron concentration and the external pressure. The system properties are then determined by the quantum critical fluctuations appearing near the instability point of the Fermi surface topology.

Pronounced −Log T Divergence in Specific Heat of Nonmetallic CeOBiS2: A Mother Phase of BiS2-Based Superconductor

The low-temperature properties of CeOBiS$_2$ single crystals are studied by electrical resistivity, magnetization, and specific heat measurements. Ce 4f -electrons are found to be in a well-localized state split by crystalline-electric-field (CEF) effects. The CEF ground state is a pure $J_z$ = $\pm$ 1/2 doublet, and excited doublets are located far above. At low temperatures in zero field, we observe pronounced $-\log T$ divergence in the specific heat, revealing the presence of quantum critical fluctuations of 4f magnetic moments near a quantum critical point (QCP). Considering that CeOBiS$_2$ is a nonmetal, this phenomenon cannot be attributed to the competition between Kondo and the Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions as in numerous f-electron-based strongly correlated metals, indicating an unconventional mechanism. We suggest that CeOBiS$_2$ is the first material found to be located at a QCP among geometrically frustrated nonmetallic magnets.