Of gyrators and non-identical anyons

Abstract

Compact scalar field theories on lattices are capable of describing a large class of many-body systems, such as interacting bosons, superconducting circuit networks, spin systems and more. We show that a generic quantum geometric many-body coupling provides Chern connections between lattice nodes, which can be understood in the electric circuit language as a type of quantum gyrator. Quantum geometry thus unlocks a direct mapping from scalar fields to anyons with fractional exchange phases, relevant for quantum error correction codes and quantum chemistry computation applications. While usual Chern-Simons-type theories have relatively local connections and a homogeneous Chern-Simons level, the gyrators can connect nonlocally and have different Chern numbers for different connections. This feature introduces the notion of non-identical anyons, i.e., excitations that do not mutually satisfy the same exchange statistics. Such non-identical exchange statistics open up a microscopic pathway to a virtually unexplored class of non-local field theories breaking the Wigner superselection rule, allowing to explore non-local communication (all-to-all qubit gates) with local control.

Figures (6)

• Figure 1: Main result of our work. (a) We consider a lattice with nodes $z$ described by compact fields $\phi_z$, where generic couplings are mediated by extra degrees of freedom described by Hamiltonians $H_{a,b,\ldots}$, summarized into a total multi-terminal Hamiltonian $H_0$. (b) By including many-body quantum fluctuations, the quantum geometric contribution of $H_0$ maps the lattice onto a network of circuit nodes coupled by gyrators with quantized gyration conductance given by a (first) Chern number. This results in anyonic excitations, one for each node, with rational fractional exchange statistics depending on the gyration conductance matrix. (c) Possible realizations of coupler systems between two nodes with nonzero Chern number, either with superconducting circuits (left) or quantum cavities (right).
• Figure 2: Important aspects regarding phase compactness. (a) A nonzero Chern number between two nodes (left) essentially maps onto a gyrator (right, denoted by Tellegen's symbol), however, with the difference that the Chern number (equivalent to the gyration conductance) must be integer quantized when defined on compact superconducting phases. Conceptually, the circuit element is a three-terminal device (one terminal to ground), where in the left symbol, we omit the explicit representation of the ground for simplicity. (b) Upon including capacitive coupling, the dynamics of a circuit with $Z$ nodes reduces to a fictitious particle moving in a magnetic field on a $Z$-dimensional torus (here shown for two dimensions). Contrary to the ordinary Landau level problem in extended space, the gauge invariance on the torus comes with a number of complications. For instance, in addition to the trivial $\mathcal{W}_{0}$ loops there are two types of nontrivial loops $\mathcal{W}_{z}$ and $\mathcal{W}_{z'}$, which cannot be controlled by the value of the magnetic field only. The phase accumulated across these extra loops is defined by gate-voltage induced offset charges.
• Figure 3: Gapping the degeneracy of the lowest Landau level by means of Josephson couplings reveals the intricate gauge properties on the torus, and the resulting nonlocal interplay between charge and flux. (a) Minimal example of a circuit with two nodes, connected by a Chern number, each additionally coupled to ground via a Josephson junction. Coupling to an external magnetic field is represented by flux shifts for each Josephson junction, $\phi_{1,2}^{\left(\text{ext}\right)}$, and interaction to a static electric field is accounted for by a capacitive coupling to a gate voltage, resulting in the offset charges $n_{1,2}^{\left(\text{ext}\right)}$. The external fields result in a finite flux and charge dispersion, which can be interpreted as a superposition of an Aharonov-Bohm, respectively, Aharonov-Casher effect. Panels (b) and (c) show the different periodicity with respect to applied flux at node 1 and applied offset charge at node 2, here for $p=C^{\left(1\right)}=3$. The broken periodicity with respect to the offset charge indicates that the anyons carry a fractional flux (but integer charge). The solid lines show the case when only one site is shunted by a Josephson junction (i.e. $\widetilde{E}_{J2}=0$), while the dashed lines mark the case when $\widetilde{E}_{J2}=\frac{1}{2} \widetilde{E}_{J1}$.
• Figure 4: Mapping from Chern networks to fermionic systems. (a) Choosing all Chern numbers identically equal 2 (in a complete graph) reduces the anyons to regular (Majorana) fermions. Quadratic and quartic interaction terms are equivalent to two-terminal, respectively 4-terminal Josephson couplings. Note that with Josephson couplings to ground (which do not conserve the total Cooper pair charge on the nodes) the circuit architecture allows for the generation of terms in the Hamiltonian which break the fermion-parity superselection rule. (b) Fermion-parity breaking implies a breaking of the no-signalling theorem, which can be probed by two participants (Alice and Bob) having access to the fermionic nodes.
• Figure 5: Discretization of the Fock space of two superconducting islands. The mesh is defined on a torus, i.e. index $N\equiv 0$.