Paul Fendley

Abstract:

We concisely review the recent evolution in the study of parafermions—exotic emergent excitations that generalize Majorana fermions and similarly underpin a host of novel phenomena. First we generalize the intimate connection between the -symmetric Ising quantum spin chain and Majorana fermions to -symmetric chains and parafermions. In particular, we highlight how parafermion chains host a topological phase featuring protected edge zero modes. We then tour several blueprints for the laboratory realization of parafermion zero modes—focusing on quantum Hall/superconductor hybrids, quantum Hall bilayers, and two-dimensional topological insulators—and describe striking experimental fingerprints that they provide. Finally, we discuss how coupled parafermion arrays in quantum Hall architectures yield topological phases that potentially furnish hardware for a universal, intrinsically decoherence-free quantum computer.

Abstract:

The theory of generalized symmetries has recently clarified how twisted sectors resolve the Callan–Rubakov paradox, where scattering of a charged particle by a magnetic monopole appeared to violate conservation laws. Here we study a more general setting of wavepackets that propagate across topological interfaces in quantum spin systems exhibiting non-invertible symmetries and across duality defects coupling dual theories. In these scenarios, we find that the transmission is always perfect and a particle traversing the interface is converted into a non-local string-like excitation. We give a systematic way of constructing such a defect by identifying its Hilbert space with the virtual bond dimension of the matrix product operator representing defect lines. Our work provides a precise characterization of topological interfaces in perfect transmission phenomena and yields a lattice analogue of the solution to the monopole paradox in quantum field theory.

Abstract:

<jats:title>Abstract</jats:title> <jats:p> Sutherland showed that the XYZ quantum spin-chain Hamiltonian commutes with the eight-vertex model transfer matrix, so that Baxter’s subsequent <jats:italic>tour de force</jats:italic> proves the integrability of both. The proof requires parametrising the Boltzmann weights using elliptic theta functions and showing they satisfy the Yang-Baxter equation. We here give a simpler derivation of the integrability of the XYZ chain by explicitly constructing an extensive sequence of conserved charges from a matrix-product operator. We show that they commute with the XYZ Hamiltonian with periodic boundary conditions or an arbitrary boundary magnetic field. A straightforward generalisation yields impurity interactions that preserve the integrability. Placing such an impurity at the edge gives an integrable generalisation of the Kondo problem with a gapped bulk. We make contact with the traditional approach by relating our matrix-product operator to products of the eight-vertex model transfer matrix. </jats:p>

Abstract:

We derive exact strong zero mode (ESZM) operators for integrable spin- S <math display="inline"> <mi>S</mi> </math> chains with open boundary conditions and a boundary field. Their locality properties are generally weaker than in the previously known cases, but they still imply infinite coherence times in the vicinity of the edges. We explain how such integrable chains possess multiple ground states describing a first-order quantum phase transition, and that the odd number of such states for integer S <math display="inline"> <mi>S</mi> </math> makes the weaker locality properties necessary. We make contact with more traditional approaches by showing how the ESZM for S=\tfrac12 <math display="inline"> <mrow> <mi>S</mi> <mo>=</mo> <mstyle displaystyle="false"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </math> acts on energy eigenstates given by solutions of the Bethe equations.