The breakdown of the celebrated Fermi liquid theory in the strange metal phase is the central enigma of correlated quantum matter. Motivated by recent experiments reporting short-lived carriers, along with the ubiquitous observations of modulated excitations in the phase diagram of cuprates, we propose a phenomenological boson-fermion model for this phase. Along with the fermions, we consider fluctuating bosons emerging from the remnants of a pair density wave as additional current carriers in the strange metal phase. These bosonic excitations are finite-momentum Cooper pairs and thus carry twice the electronic charge, and their net spin can either be zero or one arising from the two spin-
1/2 electrons. We show in a perturbative calculation within Kubo formalism that such a model displays a
TlogT dependence of the resistivity with temperature and manifests the Drude form of the ac conductivity. Furthermore, such bosons are incoherent and hence do not contribute to the Hall conductivity. Finally, the bosons emerging from finite-momentum pairs of spin-triplet symmetry also reproduce the recently observed linear-in-field magnetoresistance in the optimally doped and overdoped cuprates [P. Giraldo-Gallo et al., Science 361, 479 (2018); J. Ayres et al., Nature 595, 661 (2021)].
For High-Tc superconductors the fluctuation regime is much wider than conventional superconductors due to the short coherence length and high transition temperature. AC measurements can probe the conductivity at a wide range of frequencies, and therefore provide a test of the scaling properties predicted from theory. The measured critical exponents and width of the fluctuation peak are sample dependent, which suggests that the sample inhomogeneities may play an important role in the critical region. In this thesis we study the effects of disorder on the fluctuation conductivity in the critical regime of zero-field normal-superconducting transition by the time-dependent Ginzburg-Landau theory. We set up a discretized model of the superconductor and calculate the two-dimensional and three-dimensional scaling function without disorder above Tc. The result of the discretized model deviates slightly from previous theoretical studies of the continuous model, which can be explained by a finite short wavelength cutoff in Ginzburg-Landau theory. Our results agree well with other theoretical investigations on cutoff effects from an analytical approach. Disorder in a superconductor is modeled by a distribution of Tc's at the lattice sites. We add random disorder to a two-dimensional lattice and calculate the scaling functions averaged over 1000 disorder configurations. When disorder is weak, the scaling functions are increased when T is close Tc . At strong disorder anomalous behavior may occur at the region close to Tc.
We consider the quantum corrections to the conductivity of fermions interacting via a Chern--Simons gauge field and concentrate on the Hartree-type contributions. The first-order Hartree approximation is only valid in the limit of weak coupling $\ensuremath{\lambda}⪡{g}^{\ensuremath{-}1/2}$ to the gauge field (
g⪢1 is the dimensionless conductance) and results in an antilocalizing conductivity correction $\ensuremath{\sim}{\ensuremath{\lambda}}^{2}g\text{ }{\text{ln}}^{2}\text{ }T$. In the case of strong coupling, an infinite summation of higher-order terms is necessary, which includes both the virtual (renormalization of the frequency) and real (dephasing) processes. At intermediate temperatures,
T0⪡T⪡gT0, where ${T}_{0}\ensuremath{\sim}1/{g}^{2}\ensuremath{\tau}$ and $\ensuremath{\tau}$ is the elastic scattering time, the
T dependence of the conductivity is determined by the Hartree correction, $\ensuremath{\delta}{\ensuremath{\sigma}}^{H}(T)\ensuremath{-}\ensuremath{\delta}{\ensuremath{\sigma}}^{H}(g{T}_{0})\ensuremath{\propto}{g}^{1/2}\ensuremath{-}{(T/{T}_{0})}^{1/2}[1+\text{ln}{(g{T}_{0}/T)}^{1/2}]$, so that $\ensuremath{\sigma}(T)$ increases with lowering
T. At low temperatures,
T⪡T0, the temperature-dependent part of the Hartree correction assumes a logarithmic form with a coefficient of order unity, $\ensuremath{\delta}{\ensuremath{\sigma}}^{H}\ensuremath{\propto}\text{ln}(1/T)$. As a result, the negative exchange contribution $\ensuremath{\delta}{\ensuremath{\sigma}}^{\text{ex}}\ensuremath{\propto}\ensuremath{-}\text{ln}\text{ }g\text{ }\text{ln}(1/T)$ becomes dominant, which yields localization in the limit of $T\ensuremath{\rightarrow}0$. We further discuss dephasing at strong coupling and show that the dephasing rates are of the order of
T, owing to the interplay of inelastic scattering and renormalization. On the other hand, the dephasing length is anomalously short, ${L}_{\ensuremath{\varphi}}⪡{L}_{T}$, where
LT is the thermal length. For the case of composite fermions with long-range Coulomb interaction, the gauge-field propagator is less singular. The resulting Hartree correction has the usual sign and temperature dependence, $\ensuremath{\delta}{\ensuremath{\sigma}}^{H}\ensuremath{\propto}\text{ln}\text{ }g\text{ }\text{ln}(1/T)$, and for realistic
g is overcompensated by the negative exchange contribution due to the gauge-boson and scalar parts of the interaction. In this case, the dephasing length ${L}_{\ensuremath{\varphi}}$ is of the order of
LT for not too low temperatures and exceeds
LT for $T\ensuremath{\lesssim}g{T}_{0}$.
We study the strong localization of hard-core bosons. Using a locator expansion we find that in the insulator, unlike for typical fermion problems, nearly all low-energy scattering paths come with positive amplitudes and hence interfere constructively. As a consequence, the localization length of bosonic excitations shrinks when the constructive interference is suppressed by a magnetic field, entailing an exponentially large positive magnetoresistance, opposite to and significantly stronger than the analogous effect in fermions. Within the forward-scattering approximation, we find that the lowest-energy excitations are the most delocalized. A similar analysis applied to random field Ising models suggests that the ordering transition is due to a delocalization initiated at zero energy rather than due to the closure of a mobility gap in the paramagnet.
Motivated by neutral excitations in disordered electronic materials and systems of trapped ultracold particles with long-range interactions, we study energy-level statistics of quasiparticles with the power-law hopping Hamiltonian $\ensuremath{\propto}1/{r}^{\ensuremath{\alpha}}$ in a strong random potential. In solid-state systems such quasiparticles, which are exemplified by neutral dipolar excitations, lead to long-range correlations of local observables and may dominate energy transport. Focusing on the excitations in disordered electronic systems, we compute the energy-level correlation function ${R}_{2}(\ensuremath{\omega})$ in a finite system in the limit of sufficiently strong disorder. At small energy differences, the correlations exhibit Wigner-Dyson statistics. In particular, in the limit of very strong disorder the energy-level correlation function is given by ${R}_{2}(\ensuremath{\omega},V)={A}_{3}\frac{\ensuremath{\omega}}{{\ensuremath{\omega}}_{V}}$ for small frequencies $\ensuremath{\omega}\ensuremath{\ll}{\ensuremath{\omega}}_{V}$ and ${R}_{2}(\ensuremath{\omega},V)=1\ensuremath{-}(\ensuremath{\alpha}\ensuremath{-}d){A}_{1}{\left(\frac{{\ensuremath{\omega}}_{V}}{\ensuremath{\omega}}\right)}^{\frac{d}{\ensuremath{\alpha}}}\ensuremath{-}{A}_{2}{\left(\frac{{\ensuremath{\omega}}_{V}}{\ensuremath{\omega}}\right)}^{2}$ for large frequencies $\ensuremath{\omega}\ensuremath{\gg}{\ensuremath{\omega}}_{V}$, where ${\ensuremath{\omega}}_{V}\ensuremath{\propto}{V}^{\ensuremath{-}\frac{\ensuremath{\alpha}}{d}}$ is the characteristic matrix element of excitation hopping in a system of volume
V, and
A1,A2, and
A3 are coefficients of order unity which depend on the shape of the system. The energy-level correlation function, which we study, allows for a direct experimental observation, for example, by measuring the correlations of the ac conductance of the system at different frequencies.
An important experimental signature of localization is the low-frequency ac conductivity, which typically vanishes as ${\ensuremath{\omega}}^{\ensuremath{\phi}}$. The exponent $\ensuremath{\phi}=2$ for Anderson insulators, whereas for many body-localized insulators $\ensuremath{\phi}$ is a continuously varying exponent, $1\ensuremath{\le}\ensuremath{\phi}\ensuremath{\le}2$. In this work, we study the low-frequency ac conductivity of localized superconductors, in which disorder is strong enough to localize all quasiparticles, while remaining weak enough to leave superconductivity intact. We find that, while the ac conductivity still follows the general form $\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{\ensuremath{\phi}}$, the exponent $\ensuremath{\phi}$ can be markedly different from the characteristic value for localized insulators. This difference occurs due to singularities in the low-energy density of states, permitted by the effective particle-hole symmetry around the Fermi level. In particular, in certain symmetry classes at zero temperature, we obtain $\ensuremath{\phi}>2$. We further identify an interesting temperature-dependent crossover in the scaling form of the ac conductivity, which could be useful for the experimental characterization of localized superconductors.