How Does the Planck Scale Affect Qubits?

Gedanken experiments in quantum gravity motivate generalised uncertainty relations (GURs) implying deviations from the standard quantum statistics close to the Planck scale. These deviations have been extensively investigated for the non-spin part of the wave function, but existing models tacitly assume that spin states remain unaffected by the quantisation of the background in which the quantum matter propagates. Here, we explore a new model of nonlocal geometry in which the Planck-scale smearing of classical points generates GURs for angular momentum. These, in turn, imply an analogous generalisation of the spin uncertainty relations. The new relations correspond to a novel representation of SU(2) that acts nontrivially on both subspaces of the composite state describing matter-geometry interactions. For single particles, each spin matrix has four independent eigenvectors, corresponding to two 2-fold degenerate eigenvalues ±(ℏ+β)/2, where β is a small correction to the effective Planck’s constant. These represent the spin states of a quantum particle immersed in a quantum background geometry and the correction by β emerges as a direct result of the interaction terms. In addition to the canonical qubits states, |0⟩=|↑⟩ and |1⟩=|↓⟩, there exist two new eigenstates in which the spin of the particle becomes entangled with the spin sector of the fluctuating spacetime. We explore ways to empirically distinguish the resulting "geometric" qubits, |0′⟩ and |1′⟩, from their canonical counterparts.

Adiabatic currents for interacting fermions on a lattice

We prove an adiabatic theorem for general densities of observables that are sums of local terms in finite systems of interacting fermions, without periodicity assumptions on the Hamiltonian and with error estimates that are uniform in the size of the system. Our result provides an adiabatic expansion to all orders, in particular, also for the initial data that lie in eigenspaces of degenerate eigenvalues. Our proof is based on ideas from [ 10 ], where Bachmann et al. proved an adiabatic theorem for interacting spin systems. As one important application of this adiabatic theorem, we provide the first rigorous derivation of the adiabatic response formula for the current density induced by an adiabatic change of the Hamiltonian of a system of interacting fermions in a ground state, with error estimates uniform in the system size. We also discuss the application to quantum Hall systems.

Scaling laws and warning signs for bifurcations of SPDEs

Critical transitions (or tipping points) are drastic sudden changes observed in many dynamical systems. Large classes of critical transitions are associated with systems, which drift slowly towards a bifurcation point. In the context of stochastic ordinary differential equations, there are results on growth of variance and autocorrelation before a transition, which can be used as possible warning signs in applications. A similar theory has recently been developed in the simplest setting for stochastic partial differential equations (SPDEs) for self-adjoint operators in the drift term. This setting leads to real discrete spectrum and growth of the covariance operator via a certain scaling law. In this paper, we develop this theory substantially further. We cover the cases of complex eigenvalues, degenerate eigenvalues as well as continuous spectrum. This provides a fairly comprehensive theory for most practical applications of warning signs for SPDE bifurcations.

The Darboux transformation of the Kundu–Eckhaus equation

We construct an analytical and explicit representation of the Darboux transformation (DT) for the Kundu–Eckhaus (KE) equation. Such solution and n -fold DT T n are given in terms of determinants whose entries are expressed by the initial eigenfunctions and ‘seed’ solutions. Furthermore, the formulae for the higher order rogue wave (RW) solutions of the KE equation are also obtained by using the Taylor expansion with the use of degenerate eigenvalues λ 2 k − 1 → λ 1 = − 1 2 a + β c 2 + i c , k =1,2,3,…, all these parameters will be defined latter. These solutions have a parameter β , which denotes the strength of the non-Kerr (quintic) nonlinear and the self-frequency shift effects. We apply the contour line method to obtain analytical formulae of the length and width for the first-order RW solution of the KE equation, and then use it to study the impact of the β on the RW solution. We observe two interesting results on localization characters of β , such that if β is increasing from a /2: (i) the length of the RW solution is increasing as well, but the width is decreasing; (ii) there exist a significant rotation of the RW along the clockwise direction. We also observe the oppositely varying trend if β is increasing to a /2. We define an area of the RW solution and find that this area associated with c =1 is invariant when a and β are changing.

Graph theory and the Jahn–Teller theorem

The Jahn–Teller (JT) theorem predicts spontaneous symmetry breaking and lifting of degeneracy in degenerate electronic states of (nonlinear) molecular and solid-state systems. In these cases, degeneracy is lifted by geometric distortion. Molecular problems are often modelled using spectral theory for weighted graphs, and the present paper turns this process around and reformulates the JT theorem for general vertex- and edge-weighted graphs themselves. If the eigenvectors and eigenvalues of a general graph are considered as orbitals and energy levels (respectively) to be occupied by electrons, then degeneracy of states can be resolved by a non-totally symmetric re-weighting of edges and, where necessary, vertices. This leads to the conjecture that whenever the spectrum of a graph contains a set of bonding or anti-bonding degenerate eigenvalues, the roots of the Hamiltonian matrix over this set will show a linear dependence on edge distortions, which has the effect of lifting the degeneracy. When the degenerate level is non-bonding, distortions of vertex weights have to be included to obtain a full resolution of the eigenspace of the degeneracy. Explicit treatments are given for examples of the octahedral graph, where the degeneracy to be lifted is forced by symmetry, and the phenalenyl graph, where the degeneracy is accidental in terms of the automorphism group.