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Quantum Skyrmion Lattices

in Heisenberg Ferromagnets

Andreas Haller, Solofo Groenendijk, Alireza Habibi, Andreas Michels and Thomas L. Schmidt

Luxembourg City

Belval Campus

Outline

Introduction

Heisenberg vs. Dzyaloshinskii-Moriya vs. Zeeman

low energy configurations // phase diagram

Tensor Networks

matrix product states (MPS)

(time-dependent) variational principle

density matrix renormalization group

Quantum Spin Model

phase diagram

spin norm

entanglement

Classical Skyrmions

quasiparticles in non-centrosymmetric thin films
Heisenberg & Dzyaloshinskii-Moriya interaction
spin winding with quantized topological index
robust & long lifetimes: stable against disorder, high temperatures

Nature 442, 797 (2006) Science 323, 915 (2009) Nature 465, 901 (2010) Science 330, 1648 (2010) Journal of Applied Physics 124, 240901 (2018)

Quantum Skyrmions

quasiparticles in non-centrosymmetric thin films
Heisenberg & Dzyaloshinskii-Moriya interaction
topological witness (not quantized!)
more or less robust?
how much ''classical'' are the quantum states?

Nature Physics 7, 713 (2011) Int. J. Mod. Phys. B 33, 1930005 (2019) Phys. Rev. X 7, 041045 (2017) Phys. Rev. B 94, 134415 (2016) Phys. Rev. B 92, 245436 (2015) Phys. Rev. X 9, 041063 (2019) Phys. Rev. B 103, L060404 (2021) Phys. Rev. Lett. 127, 067201 (2021)

Classical Spin Model

\[ E = \overbrace{ J \sum_{\braket{i,j}} \bm S_{\bm r_i}\cdot\bm S_{\bm r_j} }^{\rm Heisenberg} + \overbrace{ \sum_{\braket{i,j}} \bm D_{\bm r_i,\bm r_j}\cdot\bm S_{\bm r_i}\times\bm S_{\bm r_j} }^{\rm Dzyaloshinskii-Moriya} \\ \underbrace{ +\sum_{i}\bm B_{\bm r_i}\cdot\bm S_{\bm r_i} }_{\rm external\ Zeeman\ Field} \]

Heisenberg

\[ E_J(\{\bm S_{\bm r_i},\ i=1,...,N\}) = J \sum_{\braket{i,j}} \bm S_{\bm r_i}\cdot\bm S_{\bm r_j} \]

$J > 0$

numerically find $\min_{\{\bm S_{\bm r_i},\ i=1,...,N\}}E_J$

Heisenberg

\[ E_J(\{\bm S_{\bm r_i},\ i=1,...,N\}) = J \sum_{\braket{i,j}} \bm S_{\bm r_i}\cdot\bm S_{\bm r_j} \]

$J > 0$

numerically find $\min_{\{\bm S_{\bm r_i},\ i=1,...,N\}}E_J$

Heisenberg

\[ E_J(\{\bm S_{\bm r_i},\ i=1,...,N\}) = J \sum_{\braket{i,j}} \bm S_{\bm r_i}\cdot\bm S_{\bm r_j} \]

$J < 0$

numerically find $\min_{\{\bm S_{\bm r_i},\ i=1,...,N\}}E_J$

Heisenberg

\[ E_J(\{\bm S_{\bm r_i},\ i=1,...,N\}) = J \sum_{\braket{i,j}} \bm S_{\bm r_i}\cdot\bm S_{\bm r_j} \]

$J < 0$

$J>0$

Dzyaloshinskii-Moriya (DM)

\[ E_D(\{\bm S_{\bm r_i},\ i=1,...,N\}) = \sum_{\braket{i,j}} \bm D_{\bm r_i,\bm r_j}\cdot\bm S_{\bm r_i}\times\bm S_{\bm r_j} \]

$\bm D_{\bm r_i,\bm r_j} = |D|\hat e_z\times(\bm r_j-\bm r_i)$

DM vs. Heisenberg

\[ E_{J,D} = J \sum_{\braket{i,j}} \bm S_{\bm r_i}\cdot\bm S_{\bm r_j} + \sum_{\braket{i,j}} \bm D_{\bm r_i,\bm r_j}\cdot\bm S_{\bm r_i}\times\bm S_{\bm r_j} \]

$\bm D_{\bm r_i,\bm r_j} = |D|\hat e_z\times(\bm r_j-\bm r_i)$

DM vs. Heisenberg

\[ E_{J,D} = J \sum_{\braket{i,j}} \bm S_{\bm r_i}\cdot\bm S_{\bm r_j} + \sum_{\braket{i,j}} \bm D_{\bm r_i,\bm r_j}\cdot\bm S_{\bm r_i}\times\bm S_{\bm r_j} \]

$\bm D_{\bm r_i,\bm r_j} = |D|\hat e_z\times(\bm r_j-\bm r_i)$

DM vs. Heisenberg

\[ E_{J,D} = J \sum_{\braket{i,j}} \bm S_{\bm r_i}\cdot\bm S_{\bm r_j} + \sum_{\braket{i,j}} \bm D_{\bm r_i,\bm r_j}\cdot\bm S_{\bm r_i}\times\bm S_{\bm r_j} \]

How to get a domain of locally flipped spins in 2D?

DM vs. Heisenberg vs. Zeeman

\[ E_{J,B,D} = E_{J,D} + \sum_{\bm r}\bm B_{\bm r}\cdot\bm S_{\bm r} \]

$ J = -0.5|D| $

DM vs. Heisenberg vs. Zeeman

Tensor Networks

Matrix Product States (MPS)

$\psi_{i_1,i_2,...,i_N} = {\rm tr}\left(\prod_n A^{(i_n)}_{n}\right)$

representation of generic wavefunctions
faithful simulation of ground and excited states
allows investigating dynamics or open systems

Matrix Product States (MPS)

\[ \ket{\psi} = \sum_{\bm i}\psi_{i_1,i_2,...,i_N}\ket{i_1,i_2,...,i_N}\\ = \sum_{\bm i}{\rm tr}\left(\prod_n A^{(i_n)}_{n}\right)\ket{i_1,i_2,...,i_N} \]

Matrix Product States (MPS)

\[ \ket{\psi} = \sum_{\bm i}{\rm tr}\left(\prod_n A^{(i_n)}_{n}\right)\ket{i_1,i_2,...,i_N} \]

\[ \hat O = \sum_{j,j'}O_{j',j}\ket{j'}\bra{j} \]

Matrix Product States (MPS)

\[ \hat O_{q}\ket{\psi} = {\rm tr}\left(\prod_n A^{(i_n)}_{n}\right)O_{j_q',j_q}\ket{j_q'}\braket{j_q|i_1,i_2,...,i_N}\\ = {\rm tr}\left(\prod_n A^{(i_n)}_{n}\right)O_{j_q',i_q}\ket{i_1,i_2,...,j_q',...,i_N} \]

operators change the wavefunction
operators transform the MPS matrices

Matrix Product States (MPS)

operators transform the MPS matrices
leads to matrix product operators (MPO)
efficient computation of the MPS energy

Matrix Product States (MPS)

operators transform the MPS matrices
leads to matrix product operators (MPO)
efficient computation of the MPS energy

Density matrix renormalization group

variational principle $ E_{\rm GS} = \min_{\psi}\braket{\psi|H|\psi} $
$ \mathcal L = \braket{\psi|H|\psi} - \lambda(\braket{\psi|\psi}-1) $
solve the equations of motion for each tensor $A^{(i_n)}_n$

MPS Tutorial (GitHub) Quantum Skyrmions (GitHub) Phys. Rev. Lett. 69, 2863 (1992) SciPost Phys. Lect. Notes 8 (2019) Annals of Physics 411, 167998 (2019) ITensor (Fishman,White) TeNPy (Hauschild,Pollmann)

Density matrix renormalization group

map 2D system to 1D
perform DMRG in time $T\propto\mathcal O(M^3)$
measure observables

Quantum Spin $1/2$ Model

\[ \rlap{$E$}\phantom{\hat H} = \overbrace{ J \sum_{\bm r, \bm r'} \bm S_{\bm r}\cdot\bm S_{\bm r'} }^{\rm Heisenberg} + \overbrace{ \sum_{\braket{i,j}} \bm D_{\bm r_i,\bm r_j}\cdot\bm S_{\bm r_i}\times\bm S_{\bm r_j} }^{\rm Dzyaloshinskii-Moriya} \\ \underbrace{ +\sum_{\bm r}\bm B_{\bm r}\cdot\bm S_{\bm r} }_{\rm external\ Zeeman\ Field} \quad \phantom{ \underbrace{ + K \sum_{\bm r}\hat{S}_{z,\bm r}\hat{S}_{z,\bm r} }_{\rm uniaxial\ anisotropy} } \]

Quantum Spin $1/2$ Model

\[ \hat H = \overbrace{ J \sum_{\bm r, \bm r'} \hat{\bm S}_{\bm r}\cdot\hat{\bm S}_{\bm r'} }^{\rm Heisenberg} + \overbrace{ \sum_{\braket{i,j}} \bm D_{\bm r_i,\bm r_j}\cdot\hat{\bm S}_{\bm r_i}\times\hat{\bm S}_{\bm r_j} }^{\rm Dzyaloshinskii-Moriya} \\ \underbrace{ +\sum_{\bm r}\bm B_{\bm r}\cdot\hat{\bm S}_{\bm r} }_{\rm external\ Zeeman\ Field} \phantom{ \quad \underbrace{ + K \sum_{\bm r}\hat{S}_{z,\bm r}\hat{S}_{z,\bm r} }_{\rm uniaxial\ anisotropy} } \]

Quantum Spin $1/2$ Model

\[ \hat H = \overbrace{ J \sum_{\bm r, \bm r'} \hat{\bm S}_{\bm r}\cdot\hat{\bm S}_{\bm r'} }^{\rm Heisenberg} + \overbrace{ \sum_{\braket{i,j}} \bm D_{\bm r_i,\bm r_j}\cdot\hat{\bm S}_{\bm r_i}\times\hat{\bm S}_{\bm r_j} }^{\rm Dzyaloshinskii-Moriya} \\ \underbrace{ +\sum_{\bm r}\bm B_{\bm r}\cdot\hat{\bm S}_{\bm r} }_{\rm external\ Zeeman\ Field} \quad \underbrace{ + K \sum_{\bm r}\hat{S}_{z,\bm r}\hat{S}_{z,\bm r} }_{\rm uniaxial\ anisotropy} \]

Quantum Spin Model

fix $J/|D|=-0.5$, ground states for varying $B/|D|$

Quantum Spin Model

fix $J/|D|=-0.5$, ground states for varying $B/|D|$

Quantum Spin Model

fix $J/|D|=-0.5$, ground states for varying $B/|D|$

$B=0.1|D|$

$B=0.5|D|$

$B=1.0|D|$

Quantum Spin Model

fix $J/|D|=-0.5$, ground states for varying $B/|D|$

$B = -0.1|D|$

$B = -0.5|D|$

$B = -1.0|D|$

Quantum Spin Model

fix $J/|D|=-0.5$, ground states for varying $B/|D|$

$B = -0.1|D|$

$B = -0.5|D|$

$B = -1.0|D|$

Quantum Spin Model

fix $J/|D|=-0.5$, ground states for varying $B/|D|$

$B = -0.1|D|$

$B = -0.5|D|$

$B = -1.0|D|$

Quantum Spin Model

fix $J/|D|=-0.5$, ground states for varying $B/|D|$

$B = -0.1|D|$

$B = -0.5|D|$

$B = -1.0|D|$

Quantum Spin Model

fix $J/|D|=-0.5$, ground states for varying $B/|D|$

$B = -0.1|D|$

$B = -0.5|D|$

$B = -1.0|D|$

Quantum Spin Model

fix $J/|D|=-0.5$, ground states for varying $B/|D|$

$B = -0.1|D|$

$B = -0.5|D|$

$B = -1.0|D|$

Quantum Spin Model

norm of the spin expectation value

Quantum Spin Model

average magnetization and entanglement entropy

\[ \overline m_z = \sum_n \braket{\hat S_{z,n}}/N\\ S = - {\rm tr}\hat \rho_A \ln \hat\rho_A\\ \hat\rho_A = {\rm tr}_{A^c}\hat\rho \]

Quantum Spin Model

$J=-0.5|D|$

quantum Skyrmion phase more robust against $B$

Quantum Spin Model

For spin $s=1/2$: quantum skyrmion lattices

Quantum Spin Model

For spin $s=1/2$: quantum skyrmion lattices

Quantum Spin Model

For spin $s=1/2$: quantum skyrmion lattices

Quantum Spin Model

For spin $s=1/2$: quantum skyrmion lattices

Quantum Spin Model

Concurrence: measure for pair entanglement

\[ C = |\braket{\psi|\tilde\psi}|\\ \ket{\tilde\psi} = \sigma_y\otimes\sigma_y\ket{\psi^*} \] e.g. \[ \ket{\psi} = 1/{\sqrt2}\left(\ket{\uparrow\downarrow}+\ket{\downarrow\uparrow}\right)\\ C = 1 \]

Phys. Rev. Lett. 80, 2245 (1998)

Quantum Spin Model

Concurrence: measure for pair entanglement

\[ C = |\braket{\psi|\tilde\psi}|\\ \ket{\tilde\psi} = \sigma_y\otimes\sigma_y\ket{\psi^*} \] e.g. \[ \ket{\psi} = \vphantom{\sqrt2}\ket{\uparrow\uparrow}\\ C = 0 \]

Phys. Rev. Lett. 80, 2245 (1998)

Quantum Spin Model

Concurrence: measure for pair entanglement

\[ C_{\bm r_1, \bm r_2} = {\rm max}\left\{0, \lambda_1-\lambda_2-\lambda_3-\lambda_4 \right\}\\ \lambda_i > \lambda_{i+1},\ \lambda\in\sigma(R_{\bm r_1, \bm r_2}) \] \[ R_{\bm r_1, \bm r_2} = (\sigma_y\otimes\sigma_y)\rho^*_{\bm r_1,\bm r_2}(\sigma_y\otimes\sigma_y)\\ \rho_{\bm r_1, \bm r_2,\alpha,\beta} = \braket{\hat \sigma_{\bm r_1}^\alpha\hat \sigma_{\bm r_1}^\beta} \]

Phys. Rev. Lett. 80, 2245 (1998)