A simple electronic ladder model harboring \(\mathbb{Z}_4\) parafermions

Eötvös Loránd University

Budapest University of Technology and Economics

Wigner Research Centre for Physics

Botond Osváth, Gergely Barca, Balázs Dóra,

Örs Legeza, László Oroszlány

The team

Botond Osváth

ELTE

Gergely Barcza Wigner FK

Balázs Dóra

BME

Örs Legeza

Wigner FK

Outline

What are Parafermions ? Why should you care?
Where did people look for parafermions?
Where we look for parafermions ?
Where one could look for parafermions ?

Clock models and parafermions

\displaystyle H =-J \sum_{p=1}^{L-1}\omega\hat{\alpha}_{2p}^\dagger\hat{\alpha}_{2p+1} -f \sum_{p=1}^{L} \omega\hat{\alpha}_{2p-1}^\dagger\hat{\alpha}_{2p}

\Updownarrow

\displaystyle H =-J\sum_{p=1}^{L-1}\hat{\sigma}_{p}^\dagger\hat{\sigma}_{p+1}-f\sum_{p=1}^{L}\hat{\tau}_{p}+\mathrm{h. c.}

\(f=0\rightarrow\) parafermions at the edg, \(\hat{\alpha}_1\) & \(\hat{\alpha}_{2L}\), absent form the Hamiltonian!

The missing two parafermions encode an N-fold degenerate subspace!

\sigma=\left(\begin{array}{ccc} 1 & 0 & 0\\ 0 & \Omega & 0\\ 0 & 0 & \Omega^{2} \end{array}\right)\\ \tau=\left(\begin{array}{ccc} 0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0 \end{array}\right)\\ \Omega=\text{e}^{\text{i}2\pi/N}=\omega^2

N=3 Clock model

Jordan-Wigner

\displaystyle\hat{\alpha}_{2p-1} =\hat{\sigma}_{p}\prod_{q < p}\hat{\tau}_{q} \\ \hat{\alpha}_{2p} =-\omega\hat{\tau}_{p}\hat{\sigma}_{p}\prod_{q < p}\hat{\tau}_{q}\\

Parafermion

\hat{\alpha}^N_p=\hat{1},\hat{\alpha}_p^\dagger=\hat{\alpha}^{N-1}_p

\hat{\alpha}_p\hat{\alpha}_q=\Omega^{\mathrm{sign}(q-p)}\hat{\alpha}_q\hat{\alpha}_p

Majoranas vs. parafermions

Two Majorana zero modes encode a doubly degenerate subspace.
Majorana modes can be potentially realized in non-interacting systems. (i.e. mean-field description is sufficient)
With braiding alone, Majorana modes can realize nontrivial unitary operations, but no entangling qbit gates.
Two \(\mathbb{Z}_n\) parafermions encode an \(n\)-fold degenerate subspace.
Parafermions need interaction. (i.e. mean-field description is not sufficient)
\(\mathbb{Z}_{even}\) parafermions can realize entangling gates just with braiding!
\(\mathbb{Z}_{odd}\) parafermions: route to universality

A.Hutter, D. Loss Phys. Rev. B 93, 125105 (2016)

Parafermion signatures

Robustness against disorder
Highly (>2) degenerate groundstate
Localized zero-energy excitations
Nontrivial (fractional) Josephson effect
\(\mathbb{Z}_n\rightarrow 2n\pi\) periodic

J. Alicea, P. Fendley Annu. Rev. Condens. Matter Phys. 7,119 (2016.)

Possible experimental blueprints

J. Klinovaja and D. Loss

Phys. Rev. Lett. 112, 246403 (2014)

Phys. Rev. B 90, 045118 (2014)

J. Alicea, P. Fendley Annu. Rev. Condens. Matter Phys. 7,119 (2016.)

Parafermions at TI edge

F. Zhang, C. L. Kane, Phys. Rev. Lett., 113, 036401 (2014).
C. P. Orth et al. Phys. Rev. B, 91, 081406 (2015).

J. Alicea, P. Fendley Annu. Rev. Condens. Matter Phys. 7,119 (2016.)

goal: microscopic model + DMRG

bosonised models

ITensor

Budapest DMRG

A model for 2DTI that can be digested by DMRG?

The model

two "disconnected edges" \(\zeta = {L,R}\)
explicit superconductivity and interactions
time reversal symmetry

\displaystyle H_k=\sum_{m\sigma}\left(\begin{array}{cc} c_{mL\sigma}^{\dagger} & c_{mR\sigma}^{\dagger}\end{array}\right)\left(\begin{array}{cc} -\mu & t\\ t & -\mu \end{array}\right)\left(\begin{array}{c} c_{mL\sigma}\\ c_{mR\sigma} \end{array}\right)\quad\quad\quad \quad \\ -\frac{t}{2}\sum_{m\sigma}\left[\left(\begin{array}{cc} c_{m+1L\sigma}^{\dagger} & c_{m+1R\sigma}^{\dagger}\end{array}\right)\left(\begin{array}{cc} \text{i}\sigma & 1\\ 1 & -\text{i}\sigma \end{array}\right)\left(\begin{array}{c} c_{mL\sigma}\\ c_{mR\sigma} \end{array}\right)+\text{h.c.}\right]

H=H_k+H_{sc}+H_{int}

\displaystyle H_{sc} =\sum_{m\zeta}\Delta_{m\zeta}\left[c_{m\zeta\uparrow}^{\dagger}c_{m\zeta\downarrow}^{\dagger}+\text{h.c.}\right] \quad \quad \quad \quad \quad \quad \\ H_{int}=\sum_{m\zeta}V_{m\zeta}\left [c^\dagger_{m\zeta\uparrow}c_{m\zeta\downarrow}c^\dagger_{(m+1)\zeta\uparrow}c_{(m+1)\zeta\downarrow}+\text{h.c.}\right]

H_k\approx ps_z\zeta_z

Single particle spectrum

small \(B_y\) on the left for better visibility

B_{y}\neq 0

We still have Majoranas !

B_{y}\neq 0

\Delta\neq 0

V=0

Fu, Kane Phys. Rev. Lett. 100, 096407 (2008)

Finite-size DMRG calculations: phase diagram

Finite-size DMRG calculations: excitation spectrum

\(\mu=\Delta'=0\)

Finite-size DMRG calculations: local quantities

\( \langle GS_p | n_i | GS_q \rangle \propto \delta_{pq} \)

\displaystyle H_{int}=\sum_{m\zeta}V_{m\zeta}\left [c^\dagger_{m\zeta\uparrow}c_{m\zeta\downarrow}c^\dagger_{(m+1)\zeta\uparrow}c_{(m+1)\zeta\downarrow}+\text{h.c.}\right]\\ = \sum_{m,\zeta}\frac{V_{m,\zeta}}{2}\left[ S^x_{m,\zeta}S^x_{m+1,\zeta}-S^y_{m,\zeta}S^y_{m+1,\zeta}\right]

\(\mu=\Delta'=0\), \(V/t=2.2\)

Y. Vinkler-Aviv, et al. Phys. Rev. B 96, 195421 (2017)

Finite-size DMRG calculations: Josephson spectrum

Single anisotropy term is sufficient !

isotropic exchange

anisotropic exchange

\begin{align*} H_{int}=\sum_{m,\zeta} &J \mathbf{S}_{m,\zeta}\cdot\mathbf{S}_{m+1,\zeta}\\ + &A S^x_{m,\zeta}S^x_{m+1,\zeta} \end{align*}

Parafermion signatures

Robustness against disorder ✓
Fourfold degenerate groundstate ✓
Localized zero-energy excitations ✓
Nontrivial (fractional) Josephson effect ✓
\(\mathbb{Z}_4\rightarrow 8\pi\) periodic

J. Alicea, P. Fendley Annu. Rev. Condens. Matter Phys. 7,119 (2016.)

Quantum dot arrays for parafermions!!

Hsiao et al.
Phys. Rev. X 14, 011048 (2024)

Hendrickx et al.

Nature 591, 580 (2021).

Dvir et al.

Nature 614, 445 (2023)

Bordin et al. Phys. Rev. Lett. 132, 056602 (2024)

Summary

Simple ladder model capable to capture physics at a single edge of a TI.
Interactions and superconductivity are explicitly taken into account through DMRG calculations.
Fourfold degeneracy and localized interface states can be realized.
\(8\pi\) Josephson spectrum \(\rightarrow\) parafermions!
New realization avenues for parafermions in QD arrays are suggested.

https://arxiv.org/abs/2311.07359

Phys. Rev. B 110, 085304 (2024)

Josephson spectrum.. the details

\( N_2<N_4<N_3 \)

\( N_4<N_2<N_3 \)

Local parity

P_\Omega=\prod_{p\in\Omega,\sigma}\left (-1\right )^{n_{p,\sigma}}

\Omega_1

\Omega_2

\(\mathbb{Z}_{4}\) parafermions from ordinary fermions

A. Calzona, T. Meng, M. Sassetti, T. L. Schmidt
Phys. Rev. B 98, 201110(R) (2018)

N=4 clock model/

parafermion chain

each site

has 4 states

spinful electron

in 1D wire

Hamiltonian in fermion language ...

\bar H (U,V) = H^{(2)} + U \left[ V \left(H^{(4)} + H^{(6)} \right) + (1-V) \bar H^{(4)}\right]

\displaystyle \bar H^{(4)} = -J \sum_{\sigma,j} \Big[c_{\sigma,j}^\dagger c_{\sigma, j+1} \left(-n_{-\sigma,j}-n_{-\sigma,j+1}\right) \\ + c_{\sigma,j}^\dagger c_{\sigma,j+1}^\dagger \left(n_{-\sigma,j}-n_{-\sigma,j+1}\right)\Big] +h.c.

\displaystyle \begin{aligned} H&=H^{(2)}+H^{(4)}+H^{(6)}\\ H^{(2)}&=- J \sum_{\sigma,j} \left[c_{\sigma,j}^\dagger c_{\sigma, j+1} -i \;c_{-\sigma,j}^\dagger c^\dagger_{\sigma,j+1} \right] +h.c.\,,\\ H^{(4)}&= -J\sum_{\sigma,j} \Big[ c_{\sigma,j}^\dagger c_{\sigma, j+1} \left(- n_{-\sigma,j}-n_{-\sigma,j+1}\right)\\ &+ c_{\sigma,j}^\dagger c_{-\sigma, j+1}\; i \left( n_{-\sigma,j} + n_{\sigma,j+1} \right) \\ &+ c_{-\sigma,j}^\dagger c^\dagger_{\sigma,j+1}\; i \left(n_{\sigma,j}+n_{-\sigma,j+1}\right) \\ &+ c_{\sigma,j}^\dagger c^\dagger_{\sigma,j+1} \left(n_{-\sigma,j}-n_{-\sigma,j+1}\right)\Big] + h.c.\,, \\ H^{(6)}&= - J \sum_j \Big[ - 2 i\, c_{\sigma,j}^\dagger c_{-\sigma, j+1} \left( n_{-\sigma,j} n_{\sigma,j+1}\right) \\ \qquad \qquad &- 2i \, c_{-\sigma,j}^\dagger c^\dagger_{\sigma,j+1} \left(n_{\sigma,j} n_{-\sigma,j+1}\right) \Big] +h.c. \end{aligned}

... is complicated with 3 body interactions encoded in the \(H^{(6)}\) term

Threading a flux and Josephson

\displaystyle H_0=\sum_{m=1}^{L-1}\sigma_m^\dagger\sigma_{m+1} + \mathrm{h.c.}, \quad H=H_0+C e^{i\varphi}+C^\dagger e^{-i\varphi}

\displaystyle Q=\prod_m \tau_m

parafermion charge

threading a flux method depends on the coupling...