The shape of electron wave functions determines key properties of quantum materials. Very recently, it was found that disorder can turn all wave functions into fractals (``spectrum-wide quantum criticality,'' SWQC) in models of topological and strongly correlated superconductors. The implications of SWQC for superconductivity itself remain largely unexplored. Here, the authors show that superconductivity can be strongly enhanced near the SWQC Anderson transition in two one-dimensional models. The study highlights a new potential mechanism for achieving higher transition temperatures via random or structured inhomogeneity.
The shape of electron wave functions determines key properties of quantum materials. Very recently, it was found that disorder can turn all wave functions into fractals (``spectrum-wide quantum criticality,'' SWQC) in models of topological and strongly correlated superconductors. The implications of SWQC for superconductivity itself remain largely unexplored. Here, the authors show that superconductivity can be strongly enhanced near the SWQC Anderson transition in two one-dimensional models. The study highlights a new potential mechanism for achieving higher transition temperatures via random or structured inhomogeneity.
In this thesis, we consider two topics that involve strong spatial or temporal fluctuations in interacting quantum many-body systems. We first consider far-from equilibrium quantum hydrodynamics in light-driven helical edge states of a 2D topological insulator. We then consider the enhancement of superconducting and magnetic instabilities in fermion systems with random or structured inhomogeneity, driven by the nesting of fractal wave functions. In the first part, we study quantum quenches of helical edge liquids with spin-flip inelastic scattering. Counterpropagating charge packets in helical edges can be created by an ultrashort electric pulse applied across a 2D topological insulator. Localized “hot spots” that form due to scattering enable two types of strongly nonlinear wave dynamics. First, propagating packets develop self-focusing shock fronts. Second, colliding packets with opposite charge can exhibit near-perfect retroreflection, despite strong dissipation. This leads to an interaction-mediated frequency doubling that could be detected experimentally from emitted terahertz radiation. In the second part, we study the enhanced correlation effects in many-fermion systems induced by structured or random inhomogeneities. We consider the interplay of superconductivity and a wide spectrum of critical (multifractal) wave functions (“spectrum-wide quantum criticality,” SWQC) in the one-dimensional Aubry-Andr ́e and power-law random-banded matrix models (PRBMs) with attractive interactions, using self-consistent BCS theory. We find that SWQC survives the incorporation of attractive interactions at the Anderson localization transition, whereas the pairing amplitude is maximized near this transition in both models. Although increasing inhomogeneity always depletes the superfluid stiffness, we find that superconductivity is still controlled by the fractal-enhanced amplitude near the Anderson transition. Our results suggest that SWQC, recently discovered in two-dimensional topological surface-state and nodal superconductor models, can robustly enhance superconductivity. We also study the interplay of magnetic ordering and the spectrum-wide quantum critical wave functions in PRBMs with repulsive Hubbard interaction. We employ self-consistent Hartree-Fock numerics and analytical field theory calculations using a Keldysh Finkel’stein nonlinear sigma model. We find that weak interactions delocalize the system, while strong interaction localizes the system. Ferromagnetism sets in near the Anderson transition, where the ferromagnetic spin susceptibility is also shown to peak. This indicates that ferromagnetic ordering is enhanced by the fractal wave functions. Using the sigma model, we calculate the quantum corrections to the conductance and spin susceptibility. We find that the analytical and numerical results are mutually consistent.
The topological phase transitions and localization properties in a 1D p-wave superconductor under Fibonacci quasi-periodic potential modulation are investigated in this work. By calculating the Z2 topological invariant, the topological phase diagram of the system is determined numerically. It is found that the system can transition from a topologically trivial phase to a topological Anderson superconductor phase through the Fibonacci quasi-periodic modulation. Moreover, under certain parameters, the system undergoes multiple topological Anderson superconductor phase transitions, accompanied by the emergence of zero-energy modes. However, in the case of strong disorder, the topological Anderson superconductor phase is destroyed, indicating that the topological Anderson superconductor phase can be induced only within a finite range of parameters. Furthermore, by calculating and analyzing the fractal dimension and the mean inverse participation ratio (MIPR) order parameter, the localization properties of the system are analyzed. The results show that regardless of how the disorder intensity increases, the fractal dimension values of most eigenstates always remain within a range of 0–1. Subsequently, the variations in the fractal dimensions of all eigenstates for different system sizes are studied. The results show that the fractal dimension values of most eigenstates are away from 0 and 1. These results indicate that the wavefunction in the bulk of the topological Anderson superconductor phase induced by Fibonacci quasi-periodic potential is a critical state wavefunction, with the system overall being in a critical phase. The stability of the critical phase is confirmed by scale behavior of MIPR as shown in figure. It differs from the traditional topological Anderson superconductor phase induced by random disorder or AA-type quasi-periodic disorder. The results provide new insights into and references for studying topological phase transitions and localization transitions in 1D p-wave superconductors.
We have shown that in systems where the Coulomb interaction is strongly suppressed, the superconducting transition temperature may be enhanced by disorder close to the Anderson localization transition. This phenomenon is based on the enhancement by disorder of the matrix element of attraction in the Cooper channel. For enhancement to take place one needs (i) strong disorder which makes the single-particle wave functions strongly inhomogeneous in space and (ii) strong correlation of the patterns of inhomogeneity for different wavefunctions. One case where such correlation is well known is the system close to the Anderson transition. We review the notion of multifractality of wavefunctions in this region and show how the enhancement of T c arises out of the multifractal correlations.
Topological Anderson transitions, which are direct phase transitions between topologically distinct Anderson localized phases, allow for criticality in one-dimensional disordered systems. We analyze the statistical properties of an ensemble of critical wave functions at such transitions. We find that the local moments are strongly inhomogeneous, with significant amplification towards the edges of the system. In particular, we obtain an analytic expression for the spatial profile of the local moments, which is valid at all topological Anderson transitions in one dimension, as we verify by direct comparison with numerical simulations of various lattice models.
We show that quasiperiodicity can enhance superconductivity in one-dimensional narrow-band systems with s-wave pairing. Using a generalized Aubry-André-Harper model featuring quasiperiodic modulations in both the on-site potential and the hoppings, we study superconductivity across the extended, critical, and localized phases present in the noninteracting limit. Our results show that within the critical and localized phases, the superconducting critical temperature exhibits an algebraic scaling with the interaction strength, in contrast to the conventional BCS scaling observed in periodic approximants with a similar density of states. This direct comparison establishes that the enhancement originates from the nature of the single-particle eigenstates, rather than from band flattening alone. Furthermore, we find that the superfluid weight remains finite across all phases, including the localized regime, highlighting that superconducting phase coherence is retained throughout. The effects of quasiperiodicity on the superfluid weight are most pronounced in the weak-coupling regime, indicating the nontrivial role of the localization properties of the parent eigenstates in shaping the superconducting phases.
We study the strong enhancement, induced by random hopping, of the critical temperatures characterizing the transitions to superconductivity, charge-density wave, and antiferromagnetism, which can occur in bipartite lattice models at half filling, like graphene, by means of an extended Finkel'stein nonlinear $\ensuremath{\sigma}$-model renormalization group approach. We show that if Cooper channel interaction dominates, superconducting critical temperature can be enhanced at will, since superconductivity cannot be broken by entering any Anderson insulating phase. If instead, staggered interactions are relevant, antiferromagnetic order is generated by disorder at a temperature well above that expected for a clean system.