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Register a new userThe Analytic Eigenvalue Structure of the 1+1 Dirac Oscillator
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We study the analytic structure for the eigenvalues of the one-dimensional Dirac oscillator, by analytically continuing its frequency on the complex plane. A twofold Riemann surface is found, connecting the two states of a pair of particle and antiparticle. One can, at least in principle, accomplish the transition from a positive energy state to its antiparticle state by moving the frequency continuously on the complex plane, without changing the Hamiltonian after transition. This result provides a visual explanation for the absence of a negative energy state with the quantum number n=0.
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We study the algebraic structure of the one-dimensional Dirac oscillator by extending the concept of spin symmetry to a noncommutative case. An SO(4) algebra is found connecting the eigenstates of the Dirac oscillator, in which the two elements of Cartan subalgebra are conserved quantities. Similar results are obtained in the Jaynes--Cummings model.
We study the (2+1)-dimensional Dirac oscillator in the presence of an external uniform magnetic field ($B$). We show how the change of the strength of $B$ leads to the existence of a quantum phase transition in the chirality of the system. A critical value of the strength of the external magnetic field ($B_c$) can be naturally defined in terms of physical parameters of the system. While for $B=B_c$ the fermion can be considered as a free particle without defined chirality, for $B<B_c$ ($B>B_c$) the chirality is left (right) and there exist a net potential acting on the fermion. For the three regimes defined in the quantum phase transition of chirality, we observe that the energy spectra for each regime is drastically different. Then, we consider the $z$-component of the orbital angular momentum as an order parameter that characterizes the quantum phase transition.
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The multiscale entanglement renormalization ansatz describes quantum many-body states by a hierarchical entanglement structure organized by length scale. Numerically, it has been demonstrated to capture critical lattice models and the data of the corresponding conformal field theories with high accuracy. However, a rigorous understanding of its success and precise relation to the continuum is still lacking. To address this challenge, we provide an explicit construction of entanglement-renormalization quantum circuits that rigorously approximate correlation functions of the massless Dirac conformal field theory. We directly target the continuum theory: discreteness is introduced by our choice of how to probe the system, not by any underlying short-distance lattice regulator. To achieve this, we use multiresolution analysis from wavelet theory to obtain an approximation scheme and to implement entanglement renormalization in a natural way. This could be a starting point for constructing quantum circuit approximations for more general conformal field theories.
We introduce an exact mapping between the Dirac equation in (1+1)-dimensional curved spacetime (DCS) and a multiphoton quantum Rabi model (QRM). A background of a (1+1)-dimensional black hole requires a QRM with one- and two-photon terms that can be implemented in a trapped ion for the quantum simulation of Dirac particles in curved spacetime. We illustrate our proposal with a numerical analysis of the free fall of a Dirac particle into a (1+1)-dimensional black hole, and find that the Zitterbewegung effect, measurable via the oscillatory trajectory of the Dirac particle, persists in the presence of gravity. From the duality between the squeezing term in the multiphoton QRM and the metric coupling in the DCS, we show that gravity generates squeezing of the Dirac particle wave function.
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