Continuous quantum phase transitions are widely assumed and frequently observed in various systems of quantum particles or spins. Their characteristic trait involves scaling laws governing a second-order, gradual suppression of the order parameter as the quantum critical point is approached. The localization of Cooper pairs in disordered superconductors and the resulting breakdown of superconductivity have long stood as a prototypical example. Here, we show a departure from this paradigm, showcasing that amorphous superconducting films of indium oxide undergo a distinctive, discontinuous first-order quantum phase transition tuned by disorder. Through systematic measurements of the plasmon spectrum in superconducting microwave resonators, we provide evidence for a marked jump of both the zero-temperature superfluid stiffness and the transition temperature at the critical disorder. This discontinuous transition sheds light on the previously overlooked role of repulsive interactions between Cooper pairs and the subsequent competition between superconductivity and insulating Cooper-pair glass. Furthermore, our investigation shows that the critical temperature of the films no longer relates to the pairing amplitude but aligns with the superfluid stiffness, consistent with the pseudogap regime of preformed Cooper pairs. Our findings raise fundamental new questions into the role of disorder in quantum phase transitions and carry implications for superinductances in quantum circuits.

By using the Efetov’s super-symmetric formalism we computed analytically the mean spectral density \rho(E)ρ(E) for the Lévy and the Lévy -Rosenzweig-Porter random matrices which off-diagonal elements are strongly non-Gaussian with power-law tails. This makes the standard Hubbard-Stratonovich transformation inapplicable to such problems. We used, instead, the functional Hubbard-Stratonovich transformation which allowed to solve the problem analytically for large sizes of matrices. We show that \rho(E)ρ(E) depends crucially on the control parameter that drives the system through the transition between the ergodic and the fractal phases and it can be used as an order parameter.

We calculate the conductance of a junction between a disordered superconductor and a very strong half-metallic ferromagnet admitting electrons with only one spin projection. A usual mechanism of Andreev reflection is strongly suppressed in this case since Cooper pairs are composed of electrons with opposite spins. However, this obstacle can be overcome if we take into account spin-orbit scattering inside the superconductor. Spin-orbit scattering induces a fluctuational (zero on average) spin-triplet component of the superconducting condensate, which is enough to establish Andreev transport into a strong ferromagnet. This remarkably simple mechanism is quite versatile and can explain long-range triplet proximity effect in a number of experimental setups. One particular application of the suggested effect is to measure the spin-orbit scattering time τ _SO in disordered superconducting materials. The value of Andreev conductance strongly depends on the parameter Δτ _SO and can be noticeable even in very disordered but relatively light metals like granular aluminum.

In BCS superconductors, the superfluid stiffness is virtually constant at low temperature and only slightly affected by the exponentially low density of thermal quasiparticles. Here, we present an experimental and theoretical study on the temperature dependence of superfluid stiffness $\Theta\left(T\right)$ in a strongly disordered pseudo-gaped superconductor, amorphous $\text{InO}_{x}$, which exhibits non-BCS behavior. Experimentally, we report an unusual power-law suppression of the superfluid stiffness $\delta\Theta\left(T\right)\propto T^{b}$ at $T\ll T_{c}$, with $b\sim1.6$, which we measured via the frequency shift of microwave resonators. Theoretically, by combining analytical and numerical methods to a model of a disordered superconductor with pseudogap and spatial inhomogeneities of the superconducting order parameter, we found a qualitatively similar low-temperature power-law behavior with exponent $b\sim1.6-3$ being disorder-dependent. This power-law suppression of the superfluid density occurs mainly due to the broad distribution of the superconducting order parameter that is known to exist in such superconductors [arXiv:1012.3630], even moderately far from the superconductor-insulator transition. The presence of the power-law dependence $\delta\Theta\left(T\right)\propto T^{b}$ at low $T\ll T_{c}$ demonstrates the existence of low-energy collective excitations; in turn, it implies the presence of a new channel of dissipation in inhomogeneous superconductors caused by sub-gap excitations that are not quasiparticles. Our findings have implications for the use of strongly disordered superconductors as superinductance in quantum circuits.

Recent experimental studies on strongly disordered indium oxide films have revealed an unusual first-order quantum phase transition between the superconducting and insulating states (SIT). This transition is characterized by a discontinuous jump from non-zero to zero values of superfluid stiffness at the critical point, contradicting the conventional “scaling scenario” typically associated with SIT. In this paper, we present a theoretical framework for understanding this first-order transition. Our approach is based on the concept of competition between two fundamentally distinct ground states that arise from electron pairs initially localized by strong disorder: the superconducting state and the Coulomb glass insulator. These ground states are distinguished by two crucially different order parameters, suggesting a natural expectation of a discontinuous transition between them at T=0 T=0 . This transition occurs when the magnitudes of the superconducting gap \Delta Δ and the Coulomb gap E_C EC become comparable. Additionally, we extend our analysis to low non-zero temperatures and provide a mean-field “phase diagram” in the plane of (T/\Delta,E_C/\Delta) (T/Δ,EC/Δ) . Our results reveal the existence of a natural upper bound for the kinetic inductance of strongly disordered superconductors.

We calculate the quasiparticle density of states (DoS) inside the vortex core in a granular superconductor, generalizing the classical solution applicable for dirty superconductors. A discrete version of the Usadel equation for a vortex is derived and solved numerically for a broad range of parameters. Electron DoS is found to be gapful when the coherence length \xiξ becomes comparable to the distance between neighboring grains ll. Minigap magnitude E_gEg grows from zero at \xi \approx 1.4 lξ≈1.4l to third of superconducting gap $_0 $ at \xi \approx 0.5 lξ≈0.5l. The absence of low-energy excitations is the main ingredient needed to understand strong suppression of microwave dissipation recently observed in a mixed state of granular Al.

We propose and study theoretically a new mechanism of electron-impurity scattering in doped seminconductors with large dielectric constant. It is based upon the idea of vector character of deformations caused in the crystalline lattice by any point defects siting asymmetrically in the unit cell. In result, local lattice compression due to the elastic deformations decay as 1/r2 with distance from impurity. Electron scattering (due to standard deformation potential) on such defects leads to low-temperature mobility μ(n) scaling with electron density n of the form μ(n)∝n−2/3 that is close to experimental observations on a number of relevant materials.

Theory of linear microwave response of thin films of type-II superconductors in the mixed state is developed taking into account random spatial fluctuations of the parameters of the system, such as the order parameter, diffusion coefficient, or film thickness. In the regime of collective pinning the microwave response of the system exhibits strong frequency dispersion, arising from nonequilibrium vortex core quasiparticles. The corresponding contribution to the ac conductivity is controlled by the inelastic relaxation time, and may exceed the usual Bardeen-Stephen conductivity. It is caused by the Debyetype inelastic relaxation. Debye mechanism of microwave losses may be responsible for strong effects of electromagnetic noise upon dc conductivity in the mixed state at low temperatures.

In memory of Lev Petrovich Pitaevskii, Andreev A.F., Gershtein S.S., Gurevich A.V., Dmitriev V.V., Kreines N.M., Liberman M.A., Meierovich B.E., Pokrovsky V.L., Ritus V.I., Ryutova M.P., Starobinskii A.A., Feigel’man M.V., Fomin I.A., Chaplik A.V.

We consider a non-equilibrium generalization of the mixed SYK_44+SYK_22 model and calculate the energy dissipation rate W(\omega)W(ω) that results due to periodic modulation of random quadratic matrix elements with a frequency \omegaω. We find that W(\omega)W(ω) possesses a peak at \omegaω close to the polaron energy spliting \omega_RωR found recently in [1], demonstrating physical significance of this energy scale. Next, we study the effect of energy pumping with a finite amplitude at the resonance frequency \omega_RωR and calculate, in presence of this pumping, non-equilibrium dissipation rate due to low-frequency parameteric modulation. We found unusual phenomenon similar to “dry friction” in presence of pumping.

We study an array of strongly correlated quantum dots of complex SYK type and account for the effects of quadratic terms added to the SYK Hamiltonian; both local terms and inter-dot tunneling are considered in the non-Fermi-liquid temperature range T \gg T_{FL}T≫TFL. We take into account soft-mode fluctuations and demonstrate their relevance for physical observables. Electric \sigma(\omega,p)σ(ω,p) and thermal \kappa(\omega,p)κ(ω,p) conductivities are calculated as functions of frequency and momentum for arbitrary values of the particle-hole asymmetry parameter \mathcal{E}ℰ. At low-frequencies \omega \ll Tω≪T we find the Lorenz ratio L = \kappa(0,0)/T\sigma(0,0)L=κ(0,0)/Tσ(0,0) to be non-universal and temperature-dependent. At \omega \gg Tω≫T the conductivity \sigma(\omega,p)σ(ω,p) contains a pole with nearly linear dispersion \omega \approx sp\sqrt{\ln\frac{\omega}{T}}ω≈splnωT reminiscent of the “zero-sound”, known for Fermi-liquids. We demonstrate also that the developed approach makes it possible to understand the origin of heavy Fermi liquids with anomalously large Kadowaki-Woods ratio.

We developed an analytic theory of inhomogeneous superconducting pairing in strongly disordered materials, which are moderately close to superconducting-insulator transition. Single-electron eigenstates are assumed to be Anderson-localized, with a large localization volume. Superconductivity develops due to coherent delocalization of originally localized pre-formed Cooper pairs. The key assumption of the theory is that each such pair is coupled to a large number $Z\gg1$ of similar neighboring pairs. We derived integral equations for the probability distribution $P\left(\Delta\right)$ of local superconducting order parameter $\Delta\left(\boldsymbol{r}\right)$ and analyzed their solutions in the limit of small dimensionless Cooper coupling constant $\lambda\ll1$. The shape of the order-parameter distribution is found to depend crucially upon the effective number of nearest neighbors $Z_{\text{eff}}=2\nu_{0}\Delta_{0}Z$. The solution we provide is valid both at large and small $Z_{\text{eff}}$; the latter case is nontrivial as the function $P\left(\Delta\right)$ is heavily non-Gaussian. The discovery of a broad parameter range where the distribution function $P\left(\Delta\right)$ is non-Gaussian but also non-critical (in the sense of SIT criticality) is one of our key findings. The analytic results are supplemented by numerical data, and good agreement between them is observed.

We developed a theory of electric and thermoelectric conductivity of lightly doped SrTiO3 in the non-degenerate region kBT ≥ EF , assuming that the major source of electron scattering is their interaction with soft transverse optical phonons present due to proximity to ferroelectric transition. We have used kinetic equation approach within relaxation-time approximation and we have determined energy-dependent transport relaxation time τ(E) by the iterative procedure. Using electron effective mass m and electron-transverse phonon coupling constant λ as two fitting parameters, we are able to describe quantitatively a large set of the measured temperature dependences of resistivity R(T ) and Seebeck coefficient S(T ) for a broad range of electron densities studied experimentally in recent paper [1]. In addition, we calculated Nernst ratio ν = N/B in the linear approximation over weak magnetic field in the same temperature range.

We develop a theory of superconducting pairing in low-density Strontium titanate due to quadratic coupling of electron density to soft transverse optical phonons [1]. It leads to static attractive potential between electrons which decay length leff that scales inversely with soft optical gap ωT . For low electron densities n ≤ 1018cm−3 attraction between electrons is local and transition temperature Tc was found using Ref. [2]. The Tc(n) dependence in agreement with experimental data [3] for low doping was calculated. Next, we show that suppression of Tc by hydrostatic pressure [4] and strong increase of Tc due to isotop substitution O → O observed in [5] are explained within our theory.

We develop a theory of superconducting pairing in low-density Strontium titanate due to quadratic coupling of electron density to soft transverse optical phonons. It leads to static attractive potential between electrons which decay length scales inversely with soft optical gap. For low electron densities attraction between electrons is local and transition temperature Tc was found. The Tc(n) dependence in agreement with experimental data for low doping was calculated. Next, we show that suppression of Tc by hydrostatic pressure and strong increase of Tc due to isotop substitution are explained within our theory.

We study deterministic power-law quantum hopping model with an amplitude J(r) proportional to -r(-beta) and local Gaussian disorder in low dimensions d = 1, 2 under the condition d < beta < 3d/2. We demonstrate unusual combination of exponentially decreasing density of the "tail states" and localization-delocalization transition (as function of disorder strength W) pertinent to a small (vanishing in thermodynamic limit) fraction of eigenstates. In a broad range of parameters density of states v(E) decays into the tail region E < 0 as simple exponential, v(E) = v(0)e(E/E0), while characteristic energy E-0 varies smoothly across edge localization transition. We develop simple analytic theory which describes E-0 dependence on power-law exponent beta, dimensionality d and W, and compare its predictions with exact diagonalization results. At low energies within the bare "conduction band", all eigenstates are localized due to strong quantum interference at d = 1, 2; however localization length grows fast with energy decrease, contrary to the case of usual Schrodinger equation with local disorder. (C) 2021 Elsevier Inc. All rights reserved.

We propose and investigate numerically a one-dimensional model which exhibits a non-Anderson disorder-driven transition. Such transitions have recently been attracting a great deal of attention in the context of Weyl semimetals, one-dimensional systems with long-range hopping and high-dimensional semiconductors. Our model hosts quasiparticles with the dispersion ±|k|αsignk with α<1∕2 near two points (nodes) in momentum space and includes short-range-correlated random potential which allows for scattering between the nodes and near each node. In contrast with the previously studied models in dimensions d<3, the model considered here exhibits a critical scaling of the Thouless conductance which allows for an accurate determination of the critical properties of the non-Anderson transition, with a precision significantly exceeding the results obtained from the critical scaling of the density of states, usually simulated at such transitions. We find that in the limit of the vanishing parameter ε=2α−1 the correlation-length exponent ν=2∕(3|ε|) at the transition is inconsistent with the prediction νRG=1∕|ε| of the perturbative renormalisation-group analysis. Our results allow for a numerical verification of the convergence of ε-expansions for non-Anderson disorder-driven transitions and, in general, interacting field theories near critical dimensions.

In order to understand the emergence of superconductivity it is useful to study the reverse process and identify the various pathways that lead to its destruction. One way is to increase the amount of disorder, as this leads to an increase in Coulomb repulsion that overpowers the attractive interaction responsible for Cooper pair formation. A second pathway—applicable to uniformly disordered materials—is to utilize the competition between superconductivity and Anderson localization, as this leads to electronic granularity in which phase and amplitude fluctuations of the superconducting order parameter play a role. Finally, a third pathway is to construct an array of superconducting islands coupled by some form of proximity effect that leads from a superconducting state to a state with finite resistivity, which appears like a metallic groundstate. This Review Article summarizes recent progress in understanding of these different pathways, including experiments in low dimensional materials and application in superconducting quantum devices. The breakdown of superconductivity is described as a reduction in the amplitude of the order parameter or a breakdown in phase coherence of Cooper pairs. This Review Article highlights recent results that show both mechanisms may be at play simultaneously.

We study the Sachdev-Ye-Kitaev (SYK_{4}) model with a weak SYK_{2} term of magnitude Γ beyond the simplest perturbative limit considered previously. For intermediate values of the perturbation strength, J/N≪Γ≪J/sqrt[N], fluctuations of the Schwarzian mode are suppressed, and the SYK_{4} mean-field solution remains valid beyond the timescale t_{0}∼N/J up to t_{*}∼J/Γ^{2}. The out-of-time-order correlation function displays at short time intervals exponential growth with maximal Lyapunov exponent 2πT, but its prefactor scales as T at low temperatures T≤Γ.

We develop a theory of conductivity of type-II superconductors in the flux flow regime taking into account random spatial fluctuations of the system parameters, such as the gap magnitude $\mathrm{\ensuremath{\Delta}}(\mathbf{r})$ and the diffusion coefficient $D(\mathbf{r})$. We find a contribution to the conductivity that is proportional to the inelastic relaxation time ${\ensuremath{\tau}}_{\mathrm{in}}$, which is much longer than the elastic relaxation time. This contribution is due to Debye-type relaxation, and it can be much larger than the conventional flux flow conductivity due to Bardeen and Stephen. The new contribution is expected to dominate in clean superconductors at low temperatures and in magnetic fields much smaller than ${H}_{\mathrm{c}2}$.