I discuss a one-dimensional model of interacting fermions which collective excitations are Z $_N$ -parafermions. The phase diagram of this model contains ground states with Charge Density Wave and superconducting quasi long range order. Boundaries between these phases contain zero energy parafermion modes.I discuss a one-dimensional model of interacting fermions which collective excitations are Z$_N$-parafermions. The phase diagram of this model contains ground states with Charge Density Wave and superconducting quasi long range order. Boundaries between these phases contain zero energy parafermion modes.

arXiv:1407.4002Jul 15, 20143 citations

Quantized six-vertex model at roots of unity: Frobenius property and free-parafermion spectra

Rei Inoue, Atsuo Kuniba, Yuji Terashima +1 more

We study the quantized six-vertex (Q6V) model on arbitrary admissible diagrams at $q=\varepsilon$ , where $\varepsilon$ is a primitive $N$ th root of unity with $N$ odd. For the commuting layer transfer matrices $\mathbb{T}(y)$ with mixed boundary conditions, we establish the quantum Frobenius property $\mathbb{T}(y)\mathbb{T}(\varepsilon y)\cdots \mathbb{T}(\varepsilon^{N-1}y) = \mathscr{P}_N(y^N)\mathbb{I},$ where $\mathscr{P}_N$ is a scalar Laurent polynomial obtained directly from the monomial expansion of $\mathbb{T}(y)$ . Our proof exploits the underlying three-dimensional integrability, in particular the tetrahedron equations and their consequences. The quantum Frobenius relation essentially determines the joint eigenvalues of the associated Hamiltonians up to multiplicities and shows that they take a free-parafermion form, thereby extending such structures from previously known one-dimensional examples to a broad class of genuinely two-dimensional quantized six-vertex models. The construction also admits independent local gauge choices and recovers the long-range-interacting $τ_{\; 2}$ model, which includes the original free-parafermion model as a special case. We also apply the same framework to the relativistic quantum Toda chain at odd roots of unity and obtain the free-parafermion spectra for the relevant Q6V transfer matrices with fixed boundary states.We study the quantized six-vertex (Q6V) model on arbitrary admissible diagrams at $q=\varepsilon$, where $\varepsilon$ is a primitive $N$th root of unity with $N$ odd. For the commuting layer transfer matrices $\mathbb{T}(y)$ with mixed boundary conditions, we establish the quantum Frobenius property \[ \mathbb{T}(y)\mathbb{T}(\varepsilon y)\cdots \mathbb{T}(\varepsilon^{N-1}y) = \mathscr{P}_N(y^N)\mathbb{I}, \] where $\mathscr{P}_N$ is a scalar Laurent polynomial obtained directly from the monomial expansion of $\mathbb{T}(y)$. Our proof exploits the underlying three-dimensional integrability, in particular the tetrahedron equations and their consequences. The quantum Frobenius relation essentially determines the joint eigenvalues of the associated Hamiltonians up to multiplicities and shows that they take a free-parafermion form, thereby extending such structures from previously known one-dimensional examples to a broad class of genuinely two-dimensional quantized six-vertex models. The construction also admits independent local gauge choices and recovers the long-range-interacting $τ_{\; 2}$ model, which includes the original free-parafermion model as a special case. We also apply the same framework to the relativistic quantum Toda chain at odd roots of unity and obtain the free-parafermion spectra for the relevant Q6V transfer matrices with fixed boundary states.

arXiv:2609.27740Sep 23, 20260 citations

The Semi-Infinite q-Boson System with Boundary Interaction

Jan Felipe van Diejen, Erdal Emsiz

arXiv:1308.2242Letters in Mathematical PhysicsOct 21, 20139 citations

Parafermions in a multilegged geometry: Towards a scalable parafermionic network

Udit Khanna, Moshe Goldstein, Yuval Gefen

Parafermionic zero modes are non-Abelian excitations which have been predicted to emerge at the boundary of topological phases of matter. Contrary to earlier proposals, here we show that such zero modes may also exist in multilegged star junctions of quantum Hall states. We demonstrate that the quantum states spanning the degenerate parafermionic Hilbert space may be detected and manipulated through protocols employing quantum antidots and fractional edge modes. Such star-shaped setups may be the building blocks of two-dimensional parafermionic networks.Parafermionic zero modes are non-Abelian excitations which have been predicted to emerge at the boundary of topological phases of matter. Contrary to earlier proposals, here we show that such zero modes may also exist in multilegged star junctions of quantum Hall states. We demonstrate that the quantum states spanning the degenerate parafermionic Hilbert space may be detected and manipulated through protocols employing quantum antidots and fractional edge modes. Such star-shaped setups may be the building blocks of two-dimensional parafermionic networks.

arXiv:2108.11933Physical review. B./Physical review. BApr 12, 202212 citations

_N

parafermion zero modes without Fractional Quantum Hall effect

I discuss a one-dimensional model of interacting fermions which collective excitations are Z

_N

-parafermions. The phase diagram of this model contains ground states with Charge Density Wave and superconducting quasi long range order. Boundaries between these phases contain zero energy parafermion modes.

We study the quantized six-vertex (Q6V) model on arbitrary admissible diagrams at

q=\varepsilon

, where

\varepsilon

is a primitive

N

th root of unity with

N

odd. For the commuting layer transfer matrices

\mathbb{T}(y)

with mixed boundary conditions, we establish the quantum Frobenius property

\mathbb{T}(y)\mathbb{T}(\varepsilon y)\cdots \mathbb{T}(\varepsilon^{N-1}y) = \mathscr{P}_N(y^N)\mathbb{I},

where

\mathscr{P}_N

is a scalar Laurent polynomial obtained directly from the monomial expansion of

\mathbb{T}(y)

. Our proof exploits the underlying three-dimensional integrability, in particular the tetrahedron equations and their consequences. The quantum Frobenius relation essentially determines the joint eigenvalues of the associated Hamiltonians up to multiplicities and shows that they take a free-parafermion form, thereby extending such structures from previously known one-dimensional examples to a broad class of genuinely two-dimensional quantized six-vertex models. The construction also admits independent local gauge choices and recovers the long-range-interacting

τ_{\; 2}

model, which includes the original free-parafermion model as a special case. We also apply the same framework to the relativistic quantum Toda chain at odd roots of unity and obtain the free-parafermion spectra for the relevant Q6V transfer matrices with fixed boundary states.