Topological superconductivity in the presence of correlations at the edge of topological insulators: a DMRG perspective

Department of Physics of Complex Systems, ELTE Eötvös Loránd University

MTA-BME Lendület Topology and Correlation Research Group, Budapest University of Technology and Economics

László Oroszlány

The team

Osváth Botond, ELTE

Barcza Gergely, Wigner FK

Special Thanks:

András Pályi
János Asbóth
Balázs Dóra
Miklós Werner
Zoltán Zimborás
Tibor Rakovsky
Alberto Cortijo

Outline

Majorana zero modes at TI edge and broken TRS
From Majoranas to parafermions
Microscopic model for TI edge and interactions with TRS

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

Majoranas at the edge of TIs

B_x

\Delta

B. A. Bernevig, T. L. Hughes, S.-Ch. Zhang Science, 314, 1757 (2006)

BHZ+\(B_x\)+\(\Delta\)

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

J. Alicea Rep. Prog. Phys. 75, 076501 (2012)

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

Majoranas vs. parafermions

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.
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)

\(\mathbb{Z}_{4}\) 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

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

DMRG bootcamp

S. R. White, Phys. Rev. Lett. 69, 2863 (1992)

U. Schollwoeck, Annals of Physics 326, 96 (2011)

\left|\Psi\right\rangle =\displaystyle\sum_{p_{1}p_{2}\dots p_{N}}\Psi_{p_{1}p_{2}\dots p_{N}}\left|p_{1}\right\rangle \left|p_{2}\right\rangle \cdots\left|p_{N}\right\rangle

H\left|\Psi\right\rangle =E\left|\Psi\right\rangle

MPS Ansatz!

E_{GS}=\min\frac{\left\langle \Psi\right|H\left|\Psi\right\rangle }{\left\langle \Psi\right.\left|\Psi\right\rangle }

state of the art for 1D systems

higher dimensions are tricky

\Psi_{p_1p_2\dots p_N}=\mathrm{Tr}[A_{p_1}A_{p_2}\dots A_{p_N}]

Results obtained with:

Budapest DMRG & ITensor codes

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}U_{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]

\color{gray} t

{\color{gray} -t/2}\\ {\color{orange} -it\sigma_z/2}

H_k\approx p\sigma_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

U=0

Finite size DMRG calculations: phase diagram

\mu

U\neq 0

\Delta\neq 0

Properties of the degenerate ground state

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

\langle S_z\rangle

\displaystyle \sum_{\sigma}|\langle c_{p\sigma}\rangle|^2

Phase diagram

4x deg GS !!

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

4x deg GS

2x deg GS

Conclusions and outlook

We introduced a ladder model capable to capture physics at a single edge of a TI.
DMRG calculations show that in a hybrid superconducting - interacting system fourfold degeneracy and localized interface states can be realized.
Explored different interaction terms.
The existence of parafermions in the investigated system is still inconclusive.
Characterize groundstate through Josephson periodicity (counting \(\pi\)-s).

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...