Parafermions at TRI interfaces?

Oroszlány László

Department of Physics of Complex Systems

Eötvös Loránd University, Budapest

Topological quantum computer architectures

many physical qubit = few logical qubit

degenerate subspace\(\Leftrightarrow\) topology of real space

A. Kitaev, Annals of Physics 321 2 (2006)

A. Kitaev Phys.-Usp. 44 131 (2001)

R. Lutchyn et al. Phys. Rev. Lett. 105, 077001 (2010)

Y.Oreg et al. Phys. Rev. Lett. 105, 177002 (2010)

V. Murik et al. Science 336, 1003 (2012)

topological protection

due to material properties

Majorana qubit

Kitaev "honeycomb" and "toric code"

Ising / Majorana / Topological superconductor

\displaystyle H =-J\sum_{p=1}^{L-1}\hat{\sigma}_{p}^{z}\hat{\sigma}_{p+1}^{z}-f\sum_{p=1}^{L}\hat{\sigma}_{p}^{x}

\displaystyle H =-J\sum_{p=1}^{L-1}\text{i}\hat{\gamma}_{2p}\hat{\gamma}_{2p+1}-f\sum_{p=1}^{L}\text{i}\hat{\gamma}_{2p-1}\hat{\gamma}_{2p}

\Updownarrow

\displaystyle H=-J\sum_{p=1}^{L-1}\left(\hat{a}_{p}^{\dagger}\hat{a}_{p+1}^{\dagger}+\hat{a}_{p+1}^{\dagger}\hat{a}_{p}+\text{h. c.}\right)-f\sum_{p=1}^{L}\left(-2\hat{a}_{p}^{\dagger}\hat{a}_{p}+\hat{1}\right)\\

\Updownarrow

\displaystyle\hat{\gamma}_{2p-1} =\hat{\sigma}_{p}^{z}\prod_{q < p}\hat{\sigma}_{q}^x \\ \hat{\gamma}_{2p} =\text{i}\hat{\sigma}_{p}^{x}\hat{\sigma}_{p}^{z}\prod_{q < p}\hat{\sigma}_{q}^x

\hat{\gamma}^2_k=\hat{1},\hat{\gamma}_k^\dagger=\hat{\gamma}_k

\hat{\gamma}_k\hat{\gamma}_l=-\hat{\gamma}_k\hat{\gamma}_l

Jordan-Wigner

Majorana-fermion

"standard" fermion

\hat{\gamma}_{2p-1} =\hat{a}_{p}^{\dagger}+\hat{a}_{p} \\ \hat{\gamma}_{2p} =\frac{\hat{a}_{p}^{\dagger}-\hat{a}_{p}}{\text{i}} \\ \hat{a}_{p}^{\dagger}\hat{a}_{q}+\hat{a}_{q}\hat{a}_{p}^{\dagger}=\delta_{pq}

TSC

p-wave

\(\hat{\gamma}_0\), \(\hat{\gamma}_{2L+1}\) absent!!

1D "transvers filed" Ising model

Kitaev model

Clock models and parafermions

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

\Updownarrow

\displaystyle H =-J\mathrm{e}^{\mathrm{i}\phi}\sum_{p=1}^{L-1}\hat{\sigma}_{p}^\dagger\hat{\sigma}_{p+1}-f\mathrm{e}^{\mathrm{i}\theta}\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

Majorana-ferminok vs. parafermionok

Majorana-fermionok nem "igényelnek" kölcsönhatást (átlagtér közelítés elegendő).
Majorana-fermionok mozgatásának segítségével ugyan nem triviális kvanumkapuk megvalósíthatóak, de nincs univerzális kapukészlet, e.g. összefonó kaput nem lehet realizálni a topologikus védelem feladása nélkül.
Parafermionokhoz szükséges a kölcsönhatás (átlagtér közelítésen túl kell lépni)!
\(N>2\) parafermionok segítségével megvalósítható összefonó kapu, de ez még mindig nem elegendő univerzális kvantumszámításhoz.

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

Possible experiment

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

The model

two "disconected edges" (\(\zeta = {L,R} \) )
explicit superconductivity
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)\\ -\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_k\approx p\sigma_z\zeta_z

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

Single particle spectrum

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

Finite size DMRG calculations

gap remains finite !

\(V_L=1.7,\Delta_R=0.7\)

Finite size DMRG calculations

\(Length=12,\Delta_R=0.7\)

\(V_L\)

\(E-E_{GS}\)

4x degenerate!

gap !

no gap !

Properties of the ground state

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

Interface on one side

H_{Z_4}=H_{\rm SC} + H_{W}

H_{\rm SC}= \; - \sum_{\sigma, j} t c_{\sigma, j}^{\dagger} c_{\sigma, j+1}-i \Delta c_{\bar{\sigma}, j}^{\dagger} c_{\sigma, j+1}^{\dagger} \\

\displaystyle H_{W}=-W \sum_{\sigma, j} \left [ c_{\sigma, j}^{\dagger} c_{\sigma, j+1}\left(-n_{\bar{\sigma}, j}-n_{\bar{\sigma}, j+1}\right)\right .\\ +c_{\sigma, j}^{\dagger} c_{\sigma, j+1}^{\dagger}\left(n_{\bar{\sigma}, j}-n_{\bar{\sigma}, j+1}\right) \\ + i c_{\sigma, j}^{\dagger} c_{\bar{\sigma}, j+1}\left(n_{\bar{\sigma}, j} - n_{\sigma, j+1}\right)^2 \\ \left . + i c_{\sigma, j}^{\dagger} c_{\bar{\sigma}, j+1}^{\dagger}\left( n_{\bar{\sigma}, j}- n_{\sigma, j+1}\right )^2 \right ]+ \text{H.c.}