Criticality Across the Energy Spectrum from Random, Artificial Gravitational Lensing in Two-Dimensional Dirac Superconductors

The gist: The study numerically investigates how weak, random, spatial velocity modulation in two-dimensional massless Dirac materials acts as quenched gravitational disorder (QGD), revealing that it transforms most states into a universal quantum-critical “stack” with energy-independent spatial fluctuations.

Model and Disorder Formulation

The research uses a 2D massless Dirac Hamiltonian defined by Eq. (1.1) to study the effect of QGD, where the gauge-invariant disorder strength is encoded in the quenched curvature The four velocity components couple to the spatial components of the stress tensor T ab. Five variants of velocity disorder are considered, incorporating three different local deformations of the Dirac cone: flattening or steepening, pseudospin rotations, and nematic deformation (squishing). These variants are defined based on independent random velocity components such as Independent v11(r), v22(r), v12(r), and v21(r). The physical disorder variance is expressed as λ times the square of the inverse momentum cutoff, where the disorder is formally irrelevant at the Dirac point.

Critical State Stacking Across Energy Spectrum

The study finds that critical-state stacking occurs for sufficiently weak disorder in all five models (a)–(e). This stacking improves in models (b) and (d) with increasing disorder, meaning more states become quantum-critical with identical, universal spectra throughout more of the spectrum. Model (d), which includes only pure nematic deformations of the Dirac cone, exhibits this robust behavior. In contrast, models (a) and (c), which incorporate local flattening or steepening of the cone, are most susceptible to strong disorder effects, showing supercritical (more rarified) states at low- and high-energies for increasing randomness. The generic model (e) shows intermediate behavior, with a robust energy swath with critical stacking but also supercritical behavior at low energies for stronger disorder.

Relevance to Cuprate Superconductors

The QGD model is hypothesized to arise from nodal excitations in the d-wave cuprate superconductors (SCs), due to gap inhomogeneity. The results may explain the division between low-energy “coherent” (plane-wave-like) and finite-energy “incoherent” (spatially inhomogeneous) excitations in the SC and pseudogap regimes. The model variant that best matches the cuprate phenomenology possesses quenched random pseudospin rotations and nematic fluctuations. Model (b), incorporating local rotations of the pseudospin axes relative to the coordinate axes, exhibits a phenomenology most similar to LDOS maps observed in STM studies of BSCCO.

Topological Implications and Symmetry Classes

The Dirac model with QGD resides in class DIII. The analysis suggests that the stack of critical finite-energy states found here could be associated to the thermal quantum Hall plateau transition in class D. The study contrasts these findings with conventional symmetry arguments which suggest that finite-energy states of a class DIII model should reside in the symplectic class AII. The numerical results indicate that the stacking is associated to class D rather than class AII.

Crossover Energy Scale

A key observation is the qualitative separation between local density of states (LDOS) spectra obtained below and above a weakly doping-dependent scale ∆0. For model (b), the effective energy scale ∆0 can be computed in terms of the energy cutoff Λ as a function of the dimensionless disorder strength λ. The results suggest a weak decrease of ∆0 with increasing disorder strength, which is qualitatively similar to cuprate data.

Open Questions

Future work should focus on several questions including whether the crossover scale ∆0 remains finite or vanishes in the infinite system-size L → ∞ limit. A nonzero ∆0(L → ∞) would indicate a finite-energy transition between ballistic and quantum-critical stacked behavior. Furthermore, it is questioned whether the transition from “ballistic” low-energy states to critical, finite-energy ones can explain the ubiquitous linear-in-T resistivity observed in the strange metal phase above Tc. A final question concerns whether the multifractal-stacking phenomenon enhances superconductivity in a self-consistent calculation of the gap.

Conclusion

The study uncovers a third instance of quantum-critical wave function stacking due to QGD, suggesting it might provide a simple explanation for the division between plane-wave-like and spatially inhomogeneous LDOS fluctuations measured at low and finite energies in high-Tc cuprates. The best qualitative agreement with cuprate STM phenomenology is found in model (b). Robust stacking that increases with increasing disorder strength is observed in both model (b) and model (d). The generic model (e) shows intermediate behavior, with low-energy states becoming more rarified as disorder increases. The anomalous multifractal spectra selected from an energy bin where the ratio of the DOCS to DOS is maximized for models (b), (d), and (e) are plotted for the same six disorder strengths in Figs. 10–12. The results indicate that the intrinsic disorder strength λ˜ has units of length squared. The study concludes by noting that a kink in the DOS observed at the threshold of critical stacking might arise due to quasiparticle-quasiparticle interactions.

References

[1] Criticality Across the Energy Spectrum from Random, Artificial Gravitational Lensing in Two-Dimensional Dirac Superconductors, Sayed Ali Akbar Ghorashi, Jonas F. Karcher, Seth M. Davis and Matthew S. Foster 1 Department of Physics, William & Mary, Williamsburg, Virginia 23187, USA Jonas F. Karcher Institut f¨ur NanotechnologieInstitut f¨ur QuantenMaterialien und Technologien and Institut f¨ur Theorie der Kondensierten Materie, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany Seth M. Davis Department of Physics and Astronomy, Rice University, Houston, Texas 77005 USA Matthew S. Foster Rice Center for Quantum Materials Rice University, Houston, Texas 77005 The paper is dated June 11, 2020]

[2] wave interference due to QGD transforms all but the lowest-energy states into a quantum-critical “stack” with universal, energy-independent spatial fluctuations. We study five variants of velocity disorder, incorporating three different local deformations of the Dirac cone: flattening or steepening of the cone, pseudospin rotations, and nematic deformation (squishing) of the cone

[3] QGD is encoded in the four velocity components v11(r) ≡ 1 + δv11(r), v22(r) ≡ 1 + δv22(r), v12(r), v21(r). The four potentials couple to the spatial components of the energymomentum tensor T ab All nonzero random velocity components are taken to be short-range correlated, characterized by a dimensionless disorder strength λ.

[4] QGD should arise for nodal excitations in the d-wave cuprate superconductors (SCs), due to gap inhomogeneity

[5] Understanding the interplay between strong correlations and quenched disorder in low-dimensional superconductors remains one of the key challenges in condensed matter physics [1]. While pair-breaking due to elastic impurity scattering is detrimental to superconductivity, spatial inhomogeneity can locally enhance pairing. Near a superconductor-insulator transition, the latter can occur via the spatial accumulation of Cooper pairs, due to the multifractal rarification of single-particle eigenstates [2–6]. Multifractality arises in quantumcritical states due to wave interference induced by multiple impurity scattering. Two-dimensional (2D) multifractal superconductivity has very recently come to the fore in studies of transition metal dichalcoginides [7–9].

[6] An enduring mystery is the role of spatial inhomogeneity in the high-Tc cuprate superconductors. These materials are characterized by two different energy scales in the underdoped regime: the pseudogap energy ∆1 and the smaller pairing energy ∆0; the former (latter) increases (decreases) with increased underdoping [10]. Maps of ∆1(r) obtained via STM demonstrate strong nanoscale spatial inhomogeneity that increases with underdoping. A key observation in STM scans of the lower energy scale ∆0 appears to divide fermionic excitations into two distinct classes. States with energies smaller than ∆0 behave like dispersive, “coherent” Bogoliubov-de Gennes quasiparticles, showing robust quasiparticle interference States with energies above ∆0 are instead termed “incoherent,” as they exhibit strong spatial fluctuations that vary little with energy. Renewed urgency for understanding spatial inhomogeneity comes from Ref. [17], which reported an increase in Tc with increasing disorder. Much recent work on the strange physics of the cuprates invokes gravitational descriptions of quantum criticality [18].

1. The AI system can develop a predictive model for cuprate phenomenology by using quenched gravitational disorder (QGD) as a theoretical framework to explain the division between low-energy “coherent” (plane-wave-like) and finite-energy “incoherent” (spatially inhomogeneous) excitations in the SC and pseudogap regimes.

2. The system can perform quantitative analysis of material inhomogeneity by calculating the effective energy scale ∆0 as a function of disorder strength λ, which is defined as the threshold where the DOCS becomes larger than a certain percentage of the total DOS.

3. The AI can classify topological phases and their surface states more accurately by linking observed critical phenomena to specific symmetry classes; for instance, it can predict whether finite-energy states of the class DIII model in Eq. (1.1) should reside in the symplectic class [28] or if they are associated with the thermal quantum Hall plateau transition in class D.

4. The system can differentiate between various disorder models based on their predicted physical outcomes; it can determine that isotropic flattening or steepening of the Dirac cone appear to suppress critical stacking for intermediate and strong disorder [models (a) and (c)] versus the behavior of models like (b) and (d), which exhibit a robust energy swath with critical stacking.

5. The AI can predict transport properties in strongly correlated systems by using the QGD model to suggest a mechanism for the ubiquitous linear-in-T resistivity observed in the strange metal phase above Tc, potentially linking it to the transition from “ballistic” low-energy states to critical, finite-energy ones.