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Register a new userTau-Function Theory of Quantum Chaotic Transport with beta=1,2,4
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We study the cumulants and their generating functions of the probability distributions of the conductance, shot noise and Wigner delay time in ballistic quantum dots. Our approach is based on the integrable theory of certain matrix integrals and applies to all the symmetry classes beta=1,2,4 of Random Matrix Theory. We compute the weak localization corrections to the mixed cumulants of the conductance and shot noise for beta=1,4, thus proving a number of conjectures of Khoruzhenko et al. (Phys. Rev. B, Vol. 80 (2009), 125301). We derive differential equations that characterize the cumulant generating functions for all beta=1,2,4. Furthermore, we show that the cumulant generating function of the Wigner delay time can be expressed in terms of the Painleve III transcendant. This allows us to study properties of the cumulants of the Wigner delay time in the asymptotic limit n -> infinity. Finally, for all the symmetry classes and for any number of open channels, we derive a set of recurrence relations that are very efficient for computing cumulants at all orders.
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The Painleve transcendents discovered at the turn of the XX century by pure mathematical reasoning, have later made their surprising appearance -- much in the way of Wigners miracle of appropriateness -- in various problems of theoretical physics. The notable examples include the two-dimensional Ising model, one-dimensional impenetrable Bose gas, corner and polynuclear growth models, one dimensional directed polymers, string theory, two dimensional quantum gravity, and spectral distributions of random matrices. In the present contribution, ideas of integrability are utilized to advocate emergence of an one-dimensional Toda Lattice and the fifth Painleve transcendent in the paradigmatic problem of conductance fluctuations in quantum chaotic cavities coupled to the external world via ballistic point contacts. Specifically, the cumulants of the Landauer conductance of a cavity with broken time-reversal symmetry are proven to be furnished by the coefficients of a Taylor-expanded Painleve V function. Further, the relevance of the fifth Painleve transcendent for a closely related problem of sample-to-sample fluctuations of the noise power is discussed. Finally, it is demonstrated that inclusion of tunneling effects inherent in realistic point contacts does not destroy the integrability: in this case, conductance fluctuations are shown to be governed by a two-dimensional Toda Lattice.
We consider wave propagation in a complex structure coupled to a finite number $N$ of scattering channels, such as chaotic cavities or quantum dots with external leads. Temporal aspects of the scattering process are analysed through the concept of time delays, related to the energy (or frequency) derivative of the scattering matrix $mathcal{S}$. We develop a random matrix approach to study the statistical properties of the symmetrised Wigner-Smith time-delay matrix $mathcal{Q}_s=-mathrm{i}hbar,mathcal{S}^{-1/2}big(partial_varepsilonmathcal{S}big),mathcal{S}^{-1/2}$, and obtain the joint distribution of $mathcal{S}$ and $mathcal{Q}_s$ for the system with non-ideal contacts, characterised by a finite transmission probability (per channel) $0<Tleq1$. We derive two representations of the distribution of $mathcal{Q}_s$ in terms of matrix integrals specified by the Dyson symmetry index $beta=1,2,4$ (the general case of unequally coupled channels is also discussed). We apply this to the Wigner time delay $tau_mathrm{W}=(1/N),mathrm{tr}big{mathcal{Q}_sbig}$, which is an important quantity providing the density of states of the open system. Using the obtained results, we determine the distribution $mathscr{P}_{N,beta}(tau)$ of the Wigner time delay in the weak coupling limit $NTll1$ and identify three different asymptotic regimes.
We investigate some foundational issues in the quantum theory of spin transport, in the general case when the unperturbed Hamiltonian operator $H_0$ does not commute with the spin operator in view of Rashba interactions, as in the typical models for the Quantum Spin Hall effect. A gapped periodic one-particle Hamiltonian $H_0$ is perturbed by adding a constant electric field of intensity $varepsilon ll 1$ in the $j$-th direction, and the linear response in terms of a $S$-current in the $i$-th direction is computed, where $S$ is a generalized spin operator. We derive a general formula for the spin conductivity that covers both the choice of the conventional and of the proper spin current operator. We investigate the independence of the spin conductivity from the choice of the fundamental cell (Unit Cell Consistency), and we isolate a subclass of discrete periodic models where the conventional and the proper $S$-conductivity agree, thus showing that the controversy about the choice of the spin current operator is immaterial as far as models in this class are concerned. As a consequence of the general theory, we obtain that whenever the spin is (almost) conserved, the spin conductivity is (approximately) equal to the spin-Chern number. The method relies on the characterization of a non-equilibrium almost-stationary state (NEASS), which well approximates the physical state of the system (in the sense of space-adiabatic perturbation theory) and allows moreover to compute the response of the adiabatic $S$-current as the trace per unit volume of the $S$-current operator times the NEASS. This technique can be applied in a general framework, which includes both discrete and continuum models.
Symmetry groups are projectively represented in quantum mechanics, and crystalline symmetries are fundamental in condensed matter physics. Here, we systematically present a unified theory of quantum mechanical space groups from two complementary aspects. First, we provide a decomposition form for the space-group factor systems to characterize all quantum space groups. It consists of three factors, the factor system for the translation subgroup $L$, an in-homogeneous factor system for the point group $P$, and a factor connecting $L$ and $P$. The three factors satisfy three consistency equations, which are exactly solvable and can completely exhaust all factor systems for space groups. Second, since factors systems are classified by the second cohomology group, we show the (co)homology groups for space groups can be derived from Borels equivariant (co)homology theory, which leads to an algorithm that can compute all (co)homology groups for space groups. To demonstrate the general theory, we explicitly present quantum wallpaper groups with the $mathbb{Z}_2$ gauge group. Furthermore, as a primitive application, we find the time-reversal invariant quantum space groups with inversion symmetry can lead to a novel clifford band theory, where each band is fourfold degenerate to represent certain real Clifford algebras with topologically nontrivial pinor structures over the Brillouin zone. Our work serves as a foundation for exploring quantum mechanical space groups, and can find applications in spin liquids, unconventional superconductors, and artificial lattice systems, including cold atoms, photonic and phononic crystals, and even LC electric circuit networks.
We derive new all-purpose methods that involve the Dirac Delta distribution. Some of the new methods use derivatives in the argument of the Dirac Delta. We highlight potential avenues for applications to quantum field theory and we also exhibit a connection to the problem of blurring/deblurring in signal processing. We find that blurring, which can be thought of as a result of multi-path evolution, is, in Euclidean quantum field theory without spontaneous symmetry breaking, the strong coupling dual of the usual small coupling expansion in terms of the sum over Feynman graphs.
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