128 research outputs found
Free parafermions
The spectrum of the quantum Ising chain can be found by expressing the spins
in terms of free fermions. An analogous transformation exists for clock chains
with symmetry, but is of less use because the resulting parafermionic
operators remain interacting. Nonetheless, Baxter showed that a certain
non-hermitian (but PT-symmetric) clock Hamiltonian is "free", in the sense that
the entire spectrum is found in terms of independent energy levels, with the
striking feature that there are possibilities for occupying each level.
Here I show this directly explicitly finding shift operators obeying a
generalization of the Clifford algebra. I also find higher Hamiltonians that
commute with Baxter's and prove their spectrum comes from the same set of
energy levels. This thus provides an explicit notion of a "free parafermion". A
byproduct is an elegant method for the solution of the Ising/Kitaev chain with
spatially varying couplings.Comment: 44 pages. v2: minor rewriting, added several reference
Integrable sigma models and perturbed coset models
Sigma models arise frequently in particle physics and condensed-matter
physics as low-energy effective theories. In this paper I compute the exact
free energy at any temperature in two hierarchies of integrable sigma models in
two dimensions. These theories, the SU(N)/SO(N) and O(2P)/O(P) x O(P) models,
are asymptotically free and exhibit charge fractionalization. When the
instanton coupling theta=pi, they flow to the SU(N)_1 and O(2P)_1 conformal
field theories, respectively. I also generalize the free energy computation to
massive and massless perturbations of the coset conformal field theories
SU(N)_k/SO(N)_{2k} and O(2P)_k/O(P)_k x O(P)_k.Comment: 39 pages, 6 figure
Topological phases with parafermions: theory and blueprints
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 illustrate the intimate connection
between Z_3-symmetric "spin" chains and one-dimensional parafermion lattice
models, highlighting how the latter 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.Comment: 14 pages, 4 figures; slated for Annual Reviews of Condensed Matter
Physic
Lattice supersymmetry and order-disorder coexistence in the tricritical Ising model
We introduce and analyze a quantum spin/Majorana chain with a tricritical
Ising point separating a critical phase from a gapped phase with order-disorder
coexistence. We show that supersymmetry is not only an emergent property of the
scaling limit, but manifests itself on the lattice. Namely, we find explicit
lattice expressions for the supersymmetry generators and currents. Writing the
Hamiltonian in terms of these generators allows us to find the ground states
exactly at a frustration-free coupling. These confirm the coexistence between
two (topologically) ordered ground states and a disordered one in the gapped
phase. Deforming the model by including explicit chiral symmetry breaking, we
find the phases persist up to an unusual chiral phase transition where the
supersymmetry becomes exact even on the lattice.Comment: 5+3 pages. v2: added three short appendices, including numerics
comparing various four-fermi perturbation
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