Putative Quantum Critical Point Research Articles

Signature of quantum criticality in cuprates by charge density fluctuations

The universality of the strange metal phase in many quantum materials is often attributed to the presence of a quantum critical point (QCP), a zero-temperature phase transition ruled by quantum fluctuations. In cuprates, where superconductivity hinders direct QCP observation, indirect evidence comes from the identification of fluctuations compatible with the strange metal phase. Here we show that the recently discovered charge density fluctuations (CDF) possess the right properties to be associated to a quantum phase transition. Using resonant x-ray scattering, we studied the CDF in two families of cuprate superconductors across a wide doping range (up to p = 0.22). At p* ≈ 0.19, the putative QCP, the CDF intensity peaks, and the characteristic energy Δ is minimum, marking a wedge-shaped region in the phase diagram indicative of a quantum critical behavior, albeit with anomalies. These findings strengthen the role of charge order in explaining strange metal phenomenology and provide insights into high-temperature superconductivity.

Valley-polarized nematic order in twisted moiré systems: In-plane orbital magnetism and crossover from non-Fermi liquid to Fermi liquid

The interplay between strong correlations and non-trivial topology in twisted moir\'e systems can give rise to a rich landscape of ordered states that intertwine the spin, valley, and charge degrees of freedom. In this paper, we investigate the properties of a system that displays long-range valley-polarized nematic order. Besides breaking the threefold rotational symmetry of the triangular moir\'e superlattice, this type of order also breaks twofold rotational and time-reversal symmetries, which leads to interesting properties. First, we develop a phenomenological model to describe the onset of this ordered state in twisted moir\'e systems and to explore its signatures in their thermodynamic and electronic properties. Its main manifestation is that it triggers the emergence of in-plane orbital magnetic moments oriented along high-symmetry lattice directions. We also investigate the properties of the valley-polarized nematic state at zero temperature. Due to the existence of a dangerously irrelevant coupling $\lambda$ in the six-state clock model that describes the putative valley-polarized nematic quantum critical point, the ordered state displays a pseudo-Goldstone mode. Using a two-patch model, we compute the fermionic self-energy to show that down to very low energies, the Yukawa-like coupling between the pseudo-Goldstone mode and the electronic degrees of freedom promotes the emergence of non-Fermi liquid behavior. Below a crossover energy scale $\Omega^{*}\sim\lambda^{3/2}$, however, Fermi liquid behavior is recovered. Finally, we discuss the applicability of these results to other non-trivial nematic states, such as the spin-polarized nematic phase.

Non-Fermi Liquid Behavior in the Three-Dimensional Hubbard Model

We present a numerical study on the non-Fermi liquid behavior of a three-dimensional strongly correlated system. The Hubbard model in a simple cubic lattice is simulated by the dynamical cluster approximation; in particular, the quasi-particle weight is calculated at finite dopings for a range of temperatures. By fitting the quasi-particle weight to the marginal Fermi liquid form at finite doping near the putative quantum critical point, we find evidence of a separatrix between Fermi liquid and non-Fermi liquid regions. Our results suggest that a marginal Fermi liquid and possibly a quantum critical point exist in the non-symmetry broken solution of the three-dimensional interacting electron systems. We also calculate the spectral function, close to the half-filling, and we obtain evidence of pseudogap.

Detecting Quantum Critical Points of Correlated Systems by Quantum Convolutional Neural Network Using Data from Variational Quantum Eigensolver

Machine learning has been applied to a wide variety of models, from classical statistical mechanics to quantum strongly correlated systems, for classifying phase transitions. The recently proposed quantum convolutional neural network (QCNN) provides a new framework for using quantum circuits instead of classical neural networks as the backbone of classification methods. We present the results from training the QCNN by the wavefunctions of the variational quantum eigensolver for the one-dimensional transverse field Ising model (TFIM). We demonstrate that the QCNN identifies wavefunctions corresponding to the paramagnetic and ferromagnetic phases of the TFIM with reasonable accuracy. The QCNN can be trained to predict the corresponding ‘phase’ of wavefunctions around the putative quantum critical point even though it is trained by wavefunctions far away. The paper provides a basis for exploiting the QCNN to identify the quantum critical point.

Superconductivity enhancement in polar metal regions of Sr0.95Ba0.05TiO3 and Sr0.985Ca0.015TiO3 revealed by systematic Nb doping

Two different ferroelectric materials, Sr0.95Ba0.05TiO3 and Sr0.985Ca0.015TiO3, can be turned into polar metals with broken centrosymmetry via electron doping. Systematic substitution of Nb5+ for Ti4+ has revealed that these polar metals both commonly show a simple superconducting dome with a single convex shape. Interestingly, the superconducting transition temperature Tc is enhanced more strongly in these polar metals when compared with the nonpolar matrix Sr(Ti, Nb)O3. The maximum Tc reaches 0.75 K, which is the highest reported value among the SrTiO3-based families to date. However, the Tc enhancement is unexpectedly lower within the vicinity of the putative ferroelectric quantum critical point. The enhancement then becomes much more prominent at locations further inside the dilute carrier-density region, where the screening is less effective. These results suggest that centrosymmetry breaking, i.e., the ferroelectric nature, does not kill the superconductivity. Instead, it enhances the superconductivity directly, despite the absence of strong quantum fluctuations.

Differentiated roles of Lifshitz transition on thermodynamics and superconductivity in La2-xSrxCuO4

The effect of Lifshitz transition on thermodynamics and superconductivity in hole-doped cuprates has been heavily debated but remains an open question. In particular, an observed peak of electronic specific heat is proposed to originate from fluctuations of a putative quantum critical point p* (e.g., the termination of pseudogap at zero temperature), which is close to but distinguishable from the Lifshitz transition in overdoped La-based cuprates where the Fermi surface transforms from hole-like to electron-like. Here we report an in situ angle-resolved photoemission spectroscopy study of three-dimensional Fermi surfaces in La2-xSrxCuO4 thin films (x = 0.06 to 0.35). With accurate kz dispersion quantification, the said Lifshitz transition is determined to happen within a finite range around x = 0.21. Normal state electronic specific heat, calculated from spectroscopy-derived band parameters, reveals a doping-dependent profile with a maximum at x = 0.21 that agrees with previous thermodynamic microcalorimetry measurements. The account of the specific heat maximum by underlying band structures excludes the need for additionally dominant contribution from the quantum fluctuations at p*. A d-wave superconducting gap smoothly across the Lifshitz transition demonstrates the insensitivity of superconductivity to the dramatic density of states enhancement.

Peak in the critical current density in (CaxSr1−x)3Rh4Sn13 tuned towards the structural quantum critical point

(Ca$_{x}$Sr$_{1-x}$)$_3$Rh$_4$Sn$_{13}$ is a rare system that has been shown to display an interesting interplay between structural quantum criticality and superconductivity. A putative structural quantum critical point, which is hidden beneath a broad superconducting dome, is believed to give rise to optimized superconducting properties in (Ca$_{x}$Sr$_{1-x}$)$_3$Rh$_4$Sn$_{13}$. However, the presence of the superconducting dome itself hinders the examination of the quantum critical point through electrical transport, as the transport coefficients vanish in the superconducting state. Here, we use critical current density to explore within the superconducting dome. Our measurements reveal a large enhancement of the critical current density at the zero-temperature limit when the system is tuned towards the structural quantum critical point.

Interplay of itinerant magnetism and spin-glass behavior inFexCr1−x

When suppressing the itinerant antiferromagnetism in chromium by doping with the isostructual itinerant ferromagnet iron, a dome of spin-glass behavior emerges around a putative quantum critical point at an iron concentration $x \approx 0.15$. Here, we report a comprehensive investigation of polycrystalline samples of Fe$_{x}$Cr$_{1-x}$ in the range $0.05 \leq x \leq 0.30$ using x-ray powder diffraction, magnetization, ac susceptibility, and neutron depolarization measurements, complemented by specific heat and electrical resistivity data for $x = 0.15$. Besides antiferromagnetic ($x < 0.15$) and ferromagnetic regimes ($0.15 \leq x$), we identify a dome of reentrant spin-glass behavior at low temperatures for $0.10 \leq x \leq 0.25$ that is preceded by a precursor phenomenon. Neutron depolarization indicates an increase of the size of ferromagnetic clusters with increasing $x$ and the Mydosh parameter $\phi$, inferred from the ac susceptibility, implies a crossover from cluster-glass to superparamagnetic behavior. Taken together, these findings consistently identify Fe$_{x}$Cr$_{1-x}$ as an itinerant-electron system that permits to study the evolution of spin-glass behavior of gradually varying character in unchanged crystalline environment.

Interpreting angle-dependent magnetoresistance in layered materials: Application to cuprates

The evolution of the low temperature electronic structure of the cuprate metals from the overdoped to the underdoped side has recently been addressed through Angle-Dependant Magneto-Resistance (ADMR) experiments in La$_{1.6-x}$Nd$_{0.4}$Sr$_x$CuO$_4$. The results show a striking difference between hole dopings $p = 0.24$ and $p = 0.21$ which lie on either side of a putative quantum critical point at intermediate $p$. Motivated by this, we here study the theory of ADMR in correlated layered materials, paying special attention to the role of angle dependent quasiparticle weights $Z_{\mathbf{k}}$. Such a $Z_{\mathbf{k}}$ is expected to characterize a number of popular models of the cuprate materials, particularly when underdoped. Further, in the limit of weak interlayer hopping the quasiparticle weight will affect the $c$-axis transport measured in ADMR experiments. We show that proper inclusion of the quasiparticle weight does not support an interpretation of the data in terms of a $(\pi, \pi)$ spin density wave ordered state, in agreement with the lack of direct evidence for such order. We show that a simple model of Fermi surface reconfiguring across a van Hove point captures many of the striking differences seen between $p = 0.21$ and $p = 0.24$. We comment on why such a model may be appropriate for interpreting the ADMR data, despite having a large Fermi surface at $p = 0.21$, seemingly in contradiction with other evidence for a small Fermi surafce at that doping level.

Comparison of temperature and doping dependence of elastoresistivity near a putative nematic quantum critical point

Strong electronic nematic fluctuations have been discovered near optimal doping for several families of Fe-based superconductors, motivating the search for a possible link between these fluctuations, nematic quantum criticality, and high temperature superconductivity. Here we probe a key prediction of quantum criticality, namely power-law dependence of the associated nematic susceptibility as a function of composition and temperature approaching the compositionally tuned putative quantum critical point. To probe the ‘bare’ quantum critical point requires suppression of the superconducting state, which we achieve by using large magnetic fields, up to 45 T, while performing elastoresistivity measurements to follow the nematic susceptibility. We performed these measurements for the prototypical electron-doped pnictide, Ba(Fe1−xCox)2As2, over a dense comb of dopings. We find that close to the putative quantum critical point, the elastoresistivity appears to obey power-law behavior as a function of composition over almost a decade of variation in composition. Paradoxically, however, we also find that the temperature dependence for compositions close to the critical value cannot be described by a single power law.

Quantum criticality preempted by nematicity

Exotic physics often emerges around quantum criticality in metallic systems. Here we explore the nature of topological phase transitions between 3D double-Weyl semimetals and insulators (through annihilating double-Weyl nodes with opposite chiralities) in the presence of Coulomb interactions. From renormalization-group (RG) analysis, we find a non-Fermi-liquid quantum critical point (QCP) between the double-Weyl semimetals and insulators when artificially neglecting short-range interactions. However, it is shown that this non-Fermi-liquid QCP is actually unstable against nematic ordering when short-range interactions are correctly included in the RG analysis. In other words, the putative QCP between the semimetals and insulators is preempted by the emergence of nematic phases when Coulomb interactions are present and thus double-Weyl fermions cannot directly annihilate with each other. We further discuss the possible experimental relevance of the nematicity-preempted QCP to double-Weyl candidate materials ${\text{HgCr}}_{2}{\text{Se}}_{4}$ and ${\text{SrSi}}_{2}$.

Spin dynamics in NaFeAs and NaFe0.53Cu0.47As probed by resonant inelastic x-ray scattering

The parent compounds of iron-based superconductors are magnetically-ordered bad metals, with superconductivity appearing near a putative magnetic quantum critical point. The presence of both Hubbard repulsion and Hund's coupling leads to rich physics in these multiorbital systems, and motivated descriptions of magnetism in terms of itinerant electrons or localized spins. The NaFe$_{1-x}$Cu$_x$As series consists of magnetically-ordered bad metal ($x=0$), superconducting ($x\approx0.02$) and magnetically-ordered semiconducing/insulating ($x\approx0.5$) phases, providing a platform to investigate the connection between superconductivity, magnetism and electronic correlations. Here we use X-ray absorption spectroscopy and resonant inelastic X-ray scattering to study the valence state of Fe and spin dynamics in two NaFe$_{1-x}$Cu$_x$As compounds ($x=0$ and 0.47). We find that magnetism in both compounds arises from Fe$^{2+}$ atoms, and exhibits underdamped dispersive spin waves in their respective ordered states. The dispersion of spin excitations in NaFe$_{0.53}$Cu$_{0.47}$As is consistent with being quasi-one-dimensional. Compared to NaFeAs, the band top of spin waves in NaFe$_{0.53}$Cu$_{0.47}$As is slightly softened with significantly more spectral weight of the spin excitations. Our results indicate the spin dynamics in NaFe$_{0.53}$Cu$_{0.47}$As arise from localized magnetic moments and suggest the iron-based superconductors are proximate to a correlated insulating state with localized iron moments.

Robust Fermi-Surface Morphology of CeRhIn_{5} across the Putative Field-Induced Quantum Critical Point.

We report a comprehensive de Haas-van Alphen (dHvA) study of the heavy-fermion material CeRhIn_{5} in magnetic fields up to 70T. Several dHvA frequencies gradually emerge at high fields as a result of magnetic breakdown. Among them is the thermodynamically important β_{1} branch, which has not been observed so far. Comparison of our angle-dependent dHvA spectra with those of the non-4f compound LaRhIn_{5} and with band-structure calculations evidences that the Ce 4f electrons in CeRhIn_{5} remain localized over the whole field range. This rules out any significant Fermi-surface reconstruction, either at the suggested nematic phase transition at B^{*}≈30 T or at the putative quantum critical point at B_{c}≃50 T. Our results rather demonstrate the robustness of the Fermi surface and the localized nature of the 4f electrons inside and outside of the antiferromagnetic phase.

Anisotropic superconductivity mediated by ferroelectric fluctuations in cubic systems with spin-orbit coupling

Motivated by the experimental observation that superconductivity in bulk doped SrTiO$_{3}$ is enhanced as a putative ferroelectric quantum critical point (FE-QCP) is approached, we study the pairing instability of a cubic system in which electrons exchange low-energy ferroelectric fluctuations. Instead of the gradient coupling to the lattice distortion associated with ferroelectricity, we consider a direct coupling between the electrons and the bosonic ferroelectric field that appears in the presence of spin-orbit coupling. Working in the weak-coupling regime, we find that the pairing interaction is dominated by the soft transverse optical (TO) mode, resulting in a $T_{c}$ enhancement upon approaching the FE-QCP. Focusing on even-parity states, we find that although the $s$-wave state always wins, states with higher Cooper-pair angular momentum become close competitors as the TO mode softens. We show that the cubic anisotropy of the FE fluctuations mixes the $s$-wave and $g$-wave states, resulting in a characteristic anisotropy of the gap function. The gap anisotropy behaves non-monotonically as the FE-QCP is approached: upon decreasing the TO mode frequency, the gap anisotropy first changes sign and then increases in magnitude. We discuss the possible applications of our results to the superconducting state of SrTiO$_{3}$.

Anomalous quantum criticality in the electron-doped cuprates

In the physics of condensed matter, quantum critical phenomena and unconventional superconductivity are two major themes. In electron-doped cuprates, the low critical field (HC2) allows one to study the putative quantum critical point (QCP) at low temperature and to understand its connection to the long-standing problem of the origin of the high-TC superconductivity. Here we present measurements of the low-temperature normal-state thermopower (S) of the electron-doped cuprate superconductor La2-x Ce x CuO4 (LCCO) from x = 0.11-0.19. We observe quantum critical [Formula: see text] versus [Formula: see text] behavior over an unexpectedly wide doping range x = 0.15-0.17 above the QCP (x = 0.14), with a slope that scales monotonically with the superconducting transition temperature (TC with H = 0). The presence of quantum criticality over a wide doping range provides a window on the criticality. The thermopower behavior also suggests that the critical fluctuations are linked with TC Above the superconductivity dome, at x = 0.19, a conventional Fermi-liquid [Formula: see text] behavior is found for [Formula: see text] 40 K.

Magnon excitations and quantum critical behavior of the ferromagnet U4Ru7Ge6

We present an extensive study of the ferromagnetic heavy fermion compound U$_4$Ru$_7$Ge$_6$. Measurements of electrical resistivity, specific heat and magnetic properties show that U$_4$Ru$_7$Ge$_6$ orders ferromagnetically at ambient pressure with a Curie temperature $T_{C} = 6.8 \pm 0.3$ K. The low temperature magnetic behavior of this soft ferromagnet is dominated by the excitation of gapless spin-wave modes. Our results on the transport properties of U$_4$Ru$_7$Ge$_6$ under pressures up to $2.49$ GPa suggest that U$_4$Ru$_7$Ge$_6$ has a putative ferromagnetic quantum critical point (QCP) at $P_c \approx 1.7 \pm 0.02$ GPa. In the ordered phase, ferromagnetic magnons scatter the conduction electrons and give rise to a well defined power law temperature dependence in the resistivity. The coefficient of this term is related to the spin-wave stiffness and measurements of the very low temperature resistivity allow to accompany the behavior of this quantity as the the ferromagnetic QCP is approached. We find that the spin-wave stiffness decreases with increasing pressure implying that the transition to the non-magnetic Fermi liquid state is driven by the softening of the magnons. The observed quantum critical behavior of the magnetic stiffness is consistent with the influence of disorder in our system. At quantum criticality ($P = P_c \approx 1.7 \pm 0.02$ GPa), the resistivity shows the behavior expected for an itinerant metallic system near a ferromagnetic QCP.

Magnetic phase diagram of the iron pnictides in the presence of spin-orbit coupling: Frustration between C2 and C4 magnetic phases

We investigate the impact of spin anisotropic interactions, promoted by spin-orbit coupling, on the magnetic phase diagram of the iron-based superconductors. Three distinct magnetic phases with Bragg peaks at $(\pi,0)$ and $(0,\pi)$ are possible in these systems: one $C_2$ (i.e. orthorhombic) symmetric stripe magnetic phase and two $C_4$ (i.e. tetragonal) symmetric magnetic phases. While the spin anisotropic interactions allow the magnetic moments to point in any direction in the $C_2$ phase, they restrict the possible moment orientations in the $C_4$ phases. As a result, an interesting scenario arises in which the spin anisotropic interactions favor a $C_2$ phase, but the other spin isotropic interactions favor a $C_4$ phase. We study this frustration via both mean-field and renormalization-group approaches. We find that, to lift this frustration, a rich magnetic landscape emerges well below the magnetic transition temperature, with novel $C_2$, $C_4$, and mixed $C_2$-$C_4$ phases. Near the putative magnetic quantum critical point, spin anisotropies promote a stable Gaussian fixed point in the renormalization-group flow, which is absent in the spin isotropic case, and is associated with a near-degeneracy between $C_2$ and $C_4$ phases. We argue that this frustration is the reason why most $C_4$ phases in the iron pnictides only appear inside the $C_2$ phase, and discuss additional manifestations of this frustration in the phase diagrams of these materials.

Emergent Magnetic Degeneracy in Iron Pnictides due to the Interplay between Spin-Orbit Coupling and Quantum Fluctuations.

Recent experiments in iron pnictide superconductors reveal that, as the putative magnetic quantum critical point is approached, different types of magnetic order coexist over a narrow region of the phase diagram. Although these magnetic configurations share the same wave vectors, they break distinct symmetries of the lattice. Importantly, the highest superconducting transition temperature takes place close to this proliferation of near-degenerate magnetic states. In this Letter, we employ a renormalization group calculation to show that such a behavior naturally arises due to the effects of spin-orbit coupling on the quantum magnetic fluctuations. Formally, the enhanced magnetic degeneracy near the quantum critical point is manifested as a stable Gaussian fixed point with a large basin of attraction. Implications of our findings to the superconductivity of the iron pnictides are also discussed.

Coincident onset of charge-density-wave order at a quantum critical point in underdoped YBa2Cu3Ox

The recently demonstrated x-ray scattering approach using a free electron laser with a high field pulsed magnet has opened new opportunities to explore the charge density wave (CDW) order in cuprate high temperature superconductors. Using this approach, we substantially degrade the superconductivity with magnetic fields up to 33 T to investigate the onset of CDW order in YBa$_2$Cu$_3$O$_x$ at low temperatures near a putative quantum critical point (QCP) at $p_1\sim $ 0.08 holes per Cu. We find no CDW can be detected in a sample with a doping concentration less than $p_1$. Our results indicate that the onset of the CDW ground state lies inside the zero-field superconducting dome, and broken translational symmetry is associated with the putative QCP at $p_1$

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.