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arXiv:1303.4331 · Acta Polytechnica

Non-Hermitian Star-shaped Quantum Graphs

Miloslav Znojil

Published Jan 3, 2013• 2 citations
arXivHTMLPDF

Abstract

A compact review is given, and a few new numerical results are added to the recent studies of the q-pointed one-dimensional star-shaped quantum graphs. These graphs are assumed endowedwith certain specific, manifestly non-Hermitian point interactions, localized either in the inner vertex or in all of the outer vertices and carrying, in the latter case, an interesting zero-inflow interpretation.

DOI
10.14311/1819

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A compact review is given, and a few new numerical results are added to the recent studies of the q-pointed one-dimensional star-shaped quantum graphs. These graphs are assumed endowedwith certain specific, manifestly non-Hermitian point interactions, localized either in the inner vertex or in all of the outer vertices and carrying, in the latter case, an interesting zero-inflow interpretation.
For the non-Hermitian equilateral q-pointed star-shaped quantum graphs of our previous paper (Can. J. Phys. 90, 1287 (2012) doi:10.1139/p2012-107 ) we show that because of certain dynamical aspects of the model as controlled by the external, rotation-symmetric complex Robin boundary conditions, the spectrum is obtainable in a closed asymptotic-expansion form, in principle at least. Explicit formulae up to the second order are derived for illustration, and a few comments on their consequences are added.
We recall the solvable [Formula: see text]-symmetric quantum square well on an interval of x ∈ (–L, L) := [Formula: see text] (with an α-dependent non-Hermiticity given by Robin boundary conditions) and generalize it. In essence, we just replace the support interval [Formula: see text] (reinterpreted as an equilateral two-pointed star graph with Kirchhoff matching at the vertex x = 0) with a q-pointed equilateral star graph [Formula: see text] endowed with the simplest complex-rotation-symmetric external α-dependent Robin boundary conditions. The remarkably compact form of the secular determinant is then deduced. Its analysis reveals that (i) at any integer q = 2, 3, …, there exists the same q-independent and infinite subfamily of the real energies, and (ii) at any special q = 2, 6, 10, …, there exists another, additional, q-dependent infinite subfamily of the real energies. In the spirit of the recently proposed dynamical construction of the Hilbert space of a quantum system, the physical bound-state interpretation of these eigenvalues is finally proposed.
Abstract One usually assumes that the Schrödinger operator on a thin quantum network (“fattened graph”) can be simulated by the ordinary Schrödinger operator on the corresponding one-dimensional quantum graph. On the other hand, each quantum graph can be constructed of standard elements – star graphs. We prove that a thin star-shaped quantum network which consists of a compact domain and a few straight semi-infinite quantum wires, with a two-dimensional Schrödinger operator on it, can be simulated by the corresponding solvable model in form of a one-dimensional star graph: an outer space, with an ordinary Schrödinger operator on the leads, a resonance vertex supplied with an inner space and a finite matrix in it and an appropriate boundary condition connecting the inner and outer components of elements from the domain of the model. The model is (locally) quantitatively consistent: the scattering matrix of the model on a certain spectral interval serves an approximation of the scattering matrix of the network. The role of the constructed star-graph model as a “jump-start” in analytic perturbation procedure on continuous spectrum is discussed.
We study the overscreened multi-channel Kondo (MCK) model using the recently developed unitary renormalisation group technique. Our results display the importance of ground state degeneracy in explaining various important properties like the breakdown of screening and the presence of local non-Fermi liquids (NFLs). The impurity susceptibility of the intermediate coupling fixed point Hamiltonian in the zero-bandwidth (or star graph) limit shows a power-law divergence at low temperature. Despite the absence of inter-channel coupling in the MCK fixed point Hamiltonian, the study of mutual information between any two channels shows non-zero correlation between them. A spectral flow analysis of the star graph reveals that the degenerate ground state manifold possesses topological quantum numbers. Upon disentangling the impurity spin from its partners in the star graph, we find the presence of a local Mott liquid arising from inter-channel scattering processes. The low energy effective Hamiltonian obtained upon adding a finite non-zero conduction bath dispersion to the star graph Hamiltonian for both the two and three-channel cases displays the presence of local NFLs arising from inter-channel quantum fluctuations. Specifically, we confirm the presence of a local marginal Fermi liquid in the two channel case, whose properties show logarithmic scaling at low temperature as expected. Discontinuous behaviour is observed in several measures of ground state entanglement, signalling the underlying orthogonality catastrophe associated with the degenerate ground state manifold. We extend our results to underscreened and perfectly screened MCK models through duality arguments. A study of channel anisotropy under renormalisation flow reveals a series of quantum phase transitions due to the change in ground state degeneracy. Our work thus presents a template for the study of how a degenerate ground state manifold arising from symmetry and duality properties in a multichannel quantum impurity model can lead to novel multicritical phases at intermediate coupling.