Spinon bases in supersymmetric CFTs

We present a novel way to organise the finite size spectra of a class of conformal field theories (CFT) with $\mathcal{N}=2$ or (non-linear) $\mathcal{N}=4$ superconformal symmetry. Generalising the spinon basis of the $SU(n)_1$ WZW theories, we introduce supersymmetric spinons $(\phi^-, \phi^{+})$, which form a representation of the supersymmetry algebra. In each case, we show how to construct a multi-spinon basis of the chiral CFT spectra. The multi-spinon states are labelled by a collection $\{ n_j \}$ of (discrete) momenta. The state-content for given choice of $\{ n_j \}$ is determined through a generalised exclusion principle, similar to Haldane's `motif' rules for the $SU(n)_1$ theories. In the simplest case, which is the ${\cal N}=2$ superconformal theory with central charge $c=1$, we develop an algebraic framework similar to the Yangian symmetry of the $SU(2)_1$ theory. It includes an operator $H_2$, akin to a CFT Haldane-Shastry Hamiltonian, which is diagonalised by multi-spinon states. In all cases studied, we obtain finite partition sums by capping the spinon-momenta to some finite value. For the $\mathcal{N}=2$ superconformal CFTs, this finitisation precisely leads to the so-called M$_k$ supersymmetric lattice models with characteristic order-$k$ exclusion rules on the lattice. Finitising the $c=2$ CFT with non-linear ${\cal N}=4$ superconformal symmetry similarly gives lattice model partition sums for spin-full fermions with on-site and nearest neighbour exclusion.Comment: 36 pages, 3 figure

Many-body strategies for multi-qubit gates - quantum control through Krawtchouk chain dynamics

We propose a strategy for engineering multi-qubit quantum gates. As a first step, it employs an eigengate to map states in the computational basis to eigenstates of a suitable many-body Hamiltonian. The second step employs resonant driving to enforce a transition between a single pair of eigenstates, leaving all others unchanged. The procedure is completed by mapping back to the computational basis. We demonstrate the strategy for the case of a linear array with an even number N of qubits, with specific XX+YY couplings between nearest neighbors. For this so-called Krawtchouk chain, a 2-body driving term leads to the iSWAP$_N$ gate, which we numerically test for N = 4 and 6.Comment: 10 pages, 3 figure

Defects and degeneracies in supersymmetry protected phases

We analyse a class of 1D lattice models, known as M$_k$ models, which are characterised by an order-$k$ clustering of spin-less fermions and by ${\cal N}=2$ lattice supersymmetry. Our main result is the identification of a class of (bulk or edge) defects, that are in one-to-one correspondence with so-called spin fields in a corresponding $\mathbb{Z}_k$ parafermion CFT. In the gapped regime, injecting such defects leads to ground state degeneracies that are protected by the supersymmetry. The defects, which are closely analogous to quasi-holes over the fermionic Read-Rezayi quantum Hall states, display characteristic fusion rules, which are of Ising type for $k=2$ and of Fibonacci type for $k=3$.Comment: 6 pages, 3 figures. v3: version as publishe

Quantum gates by resonantly driving many-body eigenstates, with a focus on Polychronakos' model

Accurate, nontrivial quantum operations on many qubits are experimentally challenging. As opposed to the standard approach of compiling larger unitaries into sequences of 2-qubit gates, we propose a protocol on Hamiltonian control fields which implements highly selective multi-qubit gates in a strongly-coupled many-body quantum system. We exploit the selectiveness of resonant driving to exchange only 2 out of $2^N$ eigenstates of some background Hamiltonian, and discuss a basis transformation, the eigengate, that makes this operation relevant to the computational basis. The latter has a second use as a Hahn echo which undoes the dynamical phases due to the background Hamiltonian. We find that the error of such protocols scales favourably with the gate time as $t^{-2}$, but the protocol becomes inefficient with a growing number of qubits N. The framework is numerically tested in the context of a spin chain model first described by Polychronakos, for which we show that an earlier solution method naturally gives rise to an eigengate. Our techniques could be of independent interest for the theory of driven many-body systems.Comment: 21 pages, 7 figure

Exact results for strongly-correlated fermions in 2+1 dimensions

We derive exact results for a model of strongly-interacting spinless fermions hopping on a two-dimensional lattice. By exploiting supersymmetry, we find the number and type of ground states exactly. Exploring various lattices and limits, we show how the ground states can be frustrated, quantum critical, or combine frustration with a Wigner crystal. We show that on generic lattices, the model is in an exotic ``super-frustrated'' state characterized by an extensive ground-state entropy.Comment: 4 pages, 2 figures. v2: added discussion of "super-frustrated" state; to appear in PR

SPINON BASIS FOR (sl2^)_k INTEGRABLE HIGHEST WEIGHT MODULES AND NEW CHARACTER FORMULAS

In this note we review the spinon basis for the integrable highest weight modules of sl2^ at levels k\geq1, and give the corresponding character formula. We show that our spinon basis is intimately related to the basis proposed by Foda et al. in the principal gradation of the algebra. This gives rise to new identities for the q-dimensions of the integrable modules.Comment: 9 pages, plain TeX + amssym.def, to appear in the proceedings of `Statistical Mechanics and Quantum Field Theory,' USC, May 16-21, 199

Exact Solution of an Electronic Model of Superconductivity in 1+1 Dimensions

We study a superconducting integrable model of strongly correlated electrons in 1+1 dimensions. We construct all six Bethe Ans\"atze for the model and give explicit expressions for lowest conservation laws. We also prove a lowest weight theorem for the Bethe-Ansatz states.Comment: 37 pages, using the jytex macro-packag