We present a nonperturbative calculation of all multifractal scaling exponents at strong disorder for critical wave functions of Dirac fermions interacting with a non-Abelian random vector potential in two dimensions. The results, valid for an arbitrary number of fermionic flavors, are obtained by deriving from conformal field theory an effective Gaussian model for the wave function amplitudes and mapping it to the thermodynamics of a single particle in a random potential. Our spectrum confirms that the wave functions remain delocalized in the presence of strong disorder.
We study the model of (2 + 1)-dimensional relativistic fermions in a random non-Abelian gauge potential at criticality. The exact solution shows that the operator expansion contains a conserved current- a generator of a continuous symmetry. The presence of this operator changes the operator product expansion and gives rise to logarithmic contributions to the correlation functions at the critical point. We calculate the distribution function of the local density of states in this model and find that it follows the famous log-normal law.
Using a nonperturbative functional renormalization-group approach to the two-dimensional quantum
O(N) model, we compute the low-frequency limit $\ensuremath{\omega}\ensuremath{\rightarrow}0$ of the zero-temperature conductivity in the vicinity of the quantum critical point. Our results are obtained from a derivative expansion to second order of a scale-dependent effective action in the presence of an external (i.e., nondynamical) non-Abelian gauge field. While in the disordered phase the conductivity tensor $\ensuremath{\sigma}(\ensuremath{\omega})$ is diagonal, in the ordered phase it is defined, when $N\ensuremath{\ge}3$, by two independent elements, ${\ensuremath{\sigma}}_{\mathrm{A}}(\ensuremath{\omega})$ and ${\ensuremath{\sigma}}_{\mathrm{B}}(\ensuremath{\omega})$, respectively associated to
SO(N) rotations which do and do not change the direction of the order parameter. For
N=2, the conductivity in the ordered phase reduces to a single component ${\ensuremath{\sigma}}_{\mathrm{A}}(\ensuremath{\omega})$. We show that ${lim}_{\ensuremath{\omega}\ensuremath{\rightarrow}0}\ensuremath{\sigma}(\ensuremath{\omega},\ensuremath{\delta}){\ensuremath{\sigma}}_{\mathrm{A}}(\ensuremath{\omega},\ensuremath{-}\ensuremath{\delta})/{\ensuremath{\sigma}}_{q}^{2}$ is a universal number, which we compute as a function of
N ($\ensuremath{\delta}$ measures the distance to the quantum critical point,
q is the charge, and ${\ensuremath{\sigma}}_{q}={q}^{2}/h$ the quantum of conductance). On the other hand we argue that the ratio ${\ensuremath{\sigma}}_{\mathrm{B}}(\ensuremath{\omega}\ensuremath{\rightarrow}0)/{\ensuremath{\sigma}}_{q}$ is universal in the whole ordered phase, independent of
N and, when $N\ensuremath{\rightarrow}\ensuremath{\infty}$, equal to the universal conductivity ${\ensuremath{\sigma}}^{*}/{\ensuremath{\sigma}}_{q}$ at the quantum critical point.
We study the interplay between strong correlations and incommensurability on fermions using mean field as well as exact many-body Lanczos diagonalization techniques. In a two-dimensional parameter space, mean field phase diagram of infinite system shows a critical phase with multifractal characteristics sandwiched between a Bloch-type extended and a localized phase. Exact numerical computation on finite size systems of the Kohn charge stiffness Dc, characterizing transport properties, and the charge exponent Krho, characterizing the superconducting pairing fluctuations, shows the existence of a phase with superconducting correlations. This phase may be characterized by a power law scaling of the charge stiffness constant in contrast to the Anderson localized phase where Dc scales exponentially with the size of the system. We argue that this intermediate phase may be a dressed up analog of the critical phase of the noninteracting fermions in an incommensurate potential.
We obtain an explicit expression for the multipoint energy correlations of a non-solvable two-dimensional Ising models with nearest neighbor ferromagnetic interactions plus a weak finite range interaction of strength λ, in a scaling limit in which we send the lattice spacing to zero and the temperature to the critical one. Our analysis is based on an exact mapping of the model into an interacting lattice fermionic theory, which generalizes the one originally used by Schultz, Mattis, and Lieb for the nearest neighbor Ising model. The interacting model is then analyzed by a multiscale method first proposed by Pinson and Spencer. If the lattice spacing is finite, then the correlations cannot be computed in closed form: rather, they are expressed in terms of infinite, convergent, power series in λ. In the scaling limit, these infinite expansions radically simplify and reduce to the limiting energy correlations of the integrable Ising model, up to a finite renormalization of the parameters. Explicit bounds on the speed of convergence to the scaling limit are derived.
We present a study of electric transport at high temperature in a model of strongly interacting spinless fermions without disorder. We use exact diagonalization to study the statistics of the energy eigenvalues, eigenstates, and the matrix elements of the current. These suggest that our nonrandom Hamiltonian behaves like a member of a certain ensemble of Gaussian random matrices. We calculate the conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ and examine its behavior, both in finite-size samples and as extrapolated to the thermodynamic limit. We find that $\ensuremath{\sigma}(\ensuremath{\omega})$ has a prominent nondivergent singularity at $\ensuremath{\omega}=0$ reflecting a power-law long-time tail in the current autocorrelation function that arises from nonlinear couplings between the long-wavelength diffusive modes of the energy and particle number.