# Publications by Paul Fendley

## Self-dual S3-invariant quantum chains

SciPost Physics Stichting SciPost **9** (2020) 88

We investigate the self-dual three-state quantum chain with nearest-neighbor interactions and S 3 , time-reversal, and parity symmetries. We find a rich phase diagram including gapped phases with order-disorder coexistence, integrable critical points with U(1) symmetry, and ferromagnetic and antiferromagnetic critical regions described by three-state Potts and free-boson conformal field theories respectively. We also find an unusual critical phase which appears to be described by combining two conformal field theories with distinct "Fermi velocities". The order-disorder coexistence phase has an emergent fractional supersymmetry, and we find lattice analogs of its generators.

## Large classes of quantum scarred Hamiltonians from matrix product states

Physical Review B American Physical Society **102** (2020) 85120

Motivated by the existence of exact many-body quantum scars in the Affleck-Kennedy-Lieb-Tasaki (AKLT) chain, we explore the connection between matrix product state (MPS) wave functions and many-body quantum scarred Hamiltonians. We provide a method to systematically search for and construct parent Hamiltonians with towers of exact eigenstates composed of quasiparticles on top of an MPS wave function. These exact eigenstates have low entanglement in spite of being in the middle of the spectrum, thus violating the strong eigenstate thermalization hypothesis. Using our approach, we recover the AKLT chain starting from the MPS of its ground state, and we derive the most general nearest-neighbor Hamiltonian that shares the AKLT quasiparticle tower of exact eigenstates. We further apply this formalism to other simple MPS wave functions, and derive families of Hamiltonians that exhibit AKLT-like quantum scars. As a consequence, we also construct a scar-preserving deformation that connects the AKLT chain to the integrable spin-1 pure biquadratic model. Finally, we also derive other families of Hamiltonians that exhibit types of exact quantum scars, including a U ( 1 ) -invariant perturbed Potts model.

## “Not- A”, representation symmetry-protected topological, and Potts phases in an S3 -invariant chain

Physical Review B: Condensed Matter and Materials Physics American Physical Society **101** (2020) 235108

We analyze in depth an S 3 -invariant nearest-neighbor quantum chain in the region of a U ( 1 ) -invariant self-dual multicritical point. We find four distinct proximate gapped phases. One has three-state Potts order, corresponding to topological order in a parafermionic formulation. Another has “representation” symmetry-protected topological (RSPT) order, while its dual exhibits an unusual “not- A ” order, where the spins prefer to align in two of the three directions. Within each of the four phases, we find a frustration-free point with exact ground state(s). The exact ground states in the not- A phase are product states, each an equal-amplitude sum over all states where one of the three spin states on each site is absent. Their dual, the RSPT ground state, is a matrix product state similar to that of Affleck-Kennedy-Lieb-Tasaki. A field-theory analysis shows that all transition lines are in the universality class of the critical three-state Potts model. They provide a lattice realization of a flow from a free-boson field theory to the Potts conformal field theory.

## "Not-A", representation symmetry-protected topological, and Potts phases in an S-3-invariant chain

Physical Review B American Physical Society **101** (2020) 235108

We analyze in depth an S 3 -invariant nearest-neighbor quantum chain in the region of a U ( 1 ) -invariant self-dual multicritical point. We find four distinct proximate gapped phases. One has three-state Potts order, corresponding to topological order in a parafermionic formulation. Another has “representation” symmetry-protected topological (RSPT) order, while its dual exhibits an unusual “not- A ” order, where the spins prefer to align in two of the three directions. Within each of the four phases, we find a frustration-free point with exact ground state(s). The exact ground states in the not- A phase are product states, each an equal-amplitude sum over all states where one of the three spin states on each site is absent. Their dual, the RSPT ground state, is a matrix product state similar to that of Affleck-Kennedy-Lieb-Tasaki. A field-theory analysis shows that all transition lines are in the universality class of the critical three-state Potts model. They provide a lattice realization of a flow from a free-boson field theory to the Potts conformal field theory.

## Free fermions in disguise

Journal of Physics A: Mathematical and Theoretical IOP Science **52** (2019) 335002

I solve a quantum chain whose Hamiltonian is comprised solely of local four-fermi operators by constructing free-fermion raising and lowering operators. The free-fermion operators are both non-local and highly non-linear in the local fermions. This construction yields the complete spectrum of the Hamiltonian and an associated classical transfer matrix. The spatially uniform system is gapless with dynamical critical exponent z=3/2, while staggering the couplings gives a more conventional free-fermion model with an Ising transition. The Hamiltonian is equivalent to that of a spin-1/2 chain with next-nearest-neighbour interactions, and has a supersymmetry generated by a sum of fermion trilinears. The supercharges are part of a large non-abelian symmetry algebra that results in exponentially large degeneracies. The model is integrable for either open or periodic boundary conditions but the free-fermion construction only works for the former, while for the latter the extended symmetry is broken and the degeneracies split.

## Onsager symmetries in $U(1)$ -invariant clock models

Journal of Statistical Mechanics: Theory and Experiment IOP Science **2019** (2019) 043107

We show how the Onsager algebra, used in the original solution of the two-dimensional Ising model, arises as an infinite-dimensional symmetry of certain self-dual models that also have a symmetry. We describe in detail the example of nearest-neighbour n-state clock chains whose symmetry is enhanced to . As a consequence of the Onsager-algebra symmetry, the spectrum of these models possesses degeneracies with multiplicities 2 N for positive integer N. We construct the elements of the algebra explicitly from transfer matrices built from non-fundamental representations of the quantum-group algebra . We analyse the spectra further by using both the coordinate Bethe ansatz and a functional approach, and show that the degeneracies result from special exact n-string solutions of the Bethe equations. We also find a family of commuting chiral Hamiltonians that break the degeneracies and allow an integrable interpolation between ferro- and antiferromagnets.

## John Cardy's scale-invariant journey in low dimensions: a special issue for his 70th birthday Preface

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL **51** (2018) ARTN 280301

## Lattice supersymmetry and order-disorder coexistence in the tricritical Ising model

Physical Review Letters American Physical Society **120** (2018) 206403

We introduce and analyze a quantum spin or 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 also 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.

## Prethermal strong zero modes and topological qubits

Physical Review X American Physical Society **7** (2017) 041062

We prove that quantum information encoded in some topological excitations, including certain Majorana zero modes, is protected in closed systems for a time scale exponentially long in system parameters. This protection holds even at infinite temperature. At lower temperatures, the decay time becomes even longer, with a temperature dependence controlled by an effective gap that is parametrically larger than the actual energy gap of the system. This nonequilibrium dynamical phenomenon is a form of prethermalization and occurs because of obstructions to the equilibration of edge or defect degrees of freedom with the bulk. We analyze the ramifications for ordered and topological phases in one, two, and three dimensions, with examples including Majorana and parafermionic zero modes in interacting spin chains. Our results are based on a nonperturbative analysis valid in any dimension, and they are illustrated by numerical simulations in one dimension. We discuss the implications for experiments on quantum-dot chains tuned into a regime supporting end Majorana zero modes and on trapped ion chains.

## Deconfinement transitions in a generalised XY model

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL **50** (2017) ARTN 424003

## Condensation-driven phase transitions in perturbed string nets

Physical Review B **96** (2017)

## Long coherence times for edge spins

Journal of Statistical Mechanics: Theory and Experiment **2017** (2017) 063105-063105

## Topological defects on the lattice: I. The Ising model

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL **49** (2016) ARTN 354001

## Strong zero modes and eigenstate phase transitions in the XYZ/interacting Majorana chain

Journal of Physics A: Mathematical and Theoretical **49** (2016) 30LT01-30LT01

## Topological phases with parafermions: theory and blueprints

Annual Review of Condensed Matter Physics Annual Reviews **7** (2016) 119-139

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.

## Parafermionic conformal field theory on the lattice

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL **47** (2014) ARTN 452001

## Stability of zero modes in parafermion chains

PHYSICAL REVIEW B **90** (2014) ARTN 165106

## Corner contribution to the entanglement entropy of an O(3) quantum critical point in 2+1 dimensions

JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT (2014) ARTN P06009

## Geometric mutual information at classical critical points.

Physical review letters **112** (2014) 127204-

A practical use of the entanglement entropy in a 1D quantum system is to identify the conformal field theory describing its critical behavior. It is exactly (c/3)lnℓ for an interval of length ℓ in an infinite system, where c is the central charge of the conformal field theory. Here we define the geometric mutual information, an analogous quantity for classical critical points. We compute this for 2D conformal field theories in an arbitrary geometry, and show in particular that for a rectangle cut into two rectangles, it is proportional to c. This makes it possible to extract c in classical simulations, which we demonstrate for the critical Ising and three-state Potts models.

## Universal Topological Quantum Computation from a Superconductor-Abelian Quantum Hall Heterostructure

PHYSICAL REVIEW X **4** (2014) ARTN 011036