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Register a new userPrecise Wigner-Weyl calculus for lattice models
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We propose a new version of Wigner-Weyl calculus for tight-binding lattice models. It allows to express various physical quantities through Weyl symbols of operators and Greens functions. In particular, Hall conductivity in the presence of varying and arbitrarily strong magnetic field is represented using the proposed formalism as a topological invariant.
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We discuss the non-equilibrium dynamics of condensed matter/quantum field systems in the framework of Keldysh technique. In order to deal with the inhomogeneous systems we use the Wigner-Weyl formalism. Unification of the mentioned two approaches is demonstrated on the example of Hall conductivity. We express Hall conductivity through the Wigner transformed two-point Greens functions. We demonstrate how this expression is reduced to the topological number in thermal equilibrium at zero temperature. At the same time both at finite temperature and out of equilibrium the topological invariance is lost. Moreover, Hall conductivity becomes sensitive to interaction corrections.
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.
Instanton calculations in semiclassical quantum mechanics rely on integration along trajectories which solve classical equations of motion. However in systems with higher dimensionality or complexified phase space these are rarely attainable. A prime example are spin-coherent states which are used e.g. to describe single molecule magnets (SMM). We use this example to develop instanton calculus which does not rely on explicit solutions of the classical equations of motion. Energy conservation restricts the complex phase space to a Riemann surface of complex dimension one, allowing to deform integration paths according to Cauchys integral theorem. As a result, the semiclassical actions can be evaluated without knowing actual classical paths. Furthermore we show that in many cases such actions may be solely derived from monodromy properties of the corresponding Riemann surface and residue values at its singular points. As an example, we consider quenching of tunneling processes in SMM by an applied magnetic field.
In this work, we construct an alternative formulation to the traditional Algebraic Bethe ansatz for quantum integrable models derived from a generalised rational Gaudin algebra realised in terms of a collection of spins 1/2 coupled to a single bosonic mode. The ensemble of resulting models which we call Dicke-Jaynes-Cummings- Gaudin models are particularly relevant for the description of light-matter interaction in the context of quantum optics. Having two distinct ways to write any eigenstate of these models we then combine them in order to write overlaps and form factors of local operators in terms of partition functions with domain wall boundary conditions. We also demonstrate that they can all be written in terms of determinants of matrices whose entries only depend on the eigenvalues of the conserved charges. Since these eigenvalues obey a much simpler set of quadratic Bethe equations, the resulting expressions could then offer important simplifications for the numerical treatment of these models.
We review some recent developments in the study of Gibbs and non-Gibbs properties of transformed n-vector lattice and mean-field models under various transformations. Also, some new results for the loss and recovery of the Gibbs property of planar rotor models during stochastic time evolution are presented.
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